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\ 


SMITHSONIAN  CONTRIBUTIONS  TO  KNOWLEDGE. 

202 


AN 


INVESTIGATION 


OP  THE 


ORBIT  OF  URANUS, 


WITH  GENERAL  TABLES  OF  ITS  MOTION. 


t 


BY 


SIMON  NEWCOMB, 

FE0FES80B  OF  HATUEHATICS,  DKITGD  STATES  NATT. 


[AOOEFTED    FOR    PDBI.IC  ATIOK,    FKBRDART,    1S73.] 


f 


ADVERTISEMENT. 


In  tlic  investigation  of  the  Orbit  of  Uranus  wliicli  forms  the  subject  of  the  accompanying  memoir, 
as  well  as  in  tliat  of  the  Orbit  of  Neptune  previously  published  in  the  Smithsonian  Contributions, 
a  large  amount  of  arithmetical  computation  has  been  required,  especially  in  the  reduction  and  com- 
parison of  observations.  The  cost  of  this,  in  accordance  with  the  spirit  of  the  Institution  in 
advancing  science,  has  been  defrayed  from  the  income  of  the  Smithson  fund. 

As  required  by  the  rules  of  the  Institution,  the  accompanying  memoir  was  referred  to  competent 
authority  for  examination,  and  the  persons  selected  for  this  purpose  were  Professor  J.  11.  C.  CoQiu, 
of  the  Nautical  Almanac  Office,  and  Professor  Asaph  Uall,  of  the  Naval  Observatory. 


JOSEPH  HENRY, 

Secretary  S- 1. 


Wasuinoton,  1873. 


PHn,ADRi.pnrA! 

CULLI  Na,    l>R  I  KTBIt, 

7U.'t  Ja^ne  Street. 


^ 


PEEFACE. 


The  present  work  was  undertaken  as  far  back  as  the  year  1859.  But  the  labor 
devoted  to  it  at  first  amounted  to  little  more  than  tentative  efforts  to  obtain 
numerical  data  of  sufficient  accuracy,  and  to  decide  upon  a  satisfactory  method  of 
computing  the  gfcncral  perturbations  of  the  planet.  The  elements  of  Neptune 
employed  in  the  earlier  computations  were  found  to  deviate  too  widely  from  the 
truth  to  be  used  in  computing  the  perturbations  of  Uranus  with  the  first  order  of 
accuracy,  and  it  became  necessary  to  correcit  them.  This  was  done  during  the  years 
18G-1  and  1865,  and  the  investigation  was  printed  by  the  Smithsonian  Institution 
in  the  latter  year.  It  was  then  found  that  the  adopted  elements  of  Uranus  also 
differed  too  widely  from  the  truth  to  serve  is  the  basis  of  the  Avork,  and  they  were 
provisionally  corrected  by  a  scries  of  heliocentric  longitudes  derived  from  observa- 
tions extending  from  1781  to  1861.  FinalK'  it  was  found  that  the  adopted  method 
of  computing  the  perturbations,  that  of  the  "  variation  of  elements,"  though  not 
deserving  of  the  disfavor  into  which  it  has  fallen  of  late  years,  was  practically 
inapplicable  to  the  computation  of  the  most  difficult  terms,  namely,  those  of  the 
second  order  with  respccc  to  the  disturbing  forces.  Indeed,  it  appeared  to  the 
author  that  the  only  method  of  computing  these  terms  which  was  at  tiic  same  time 
general,  practicable,  and  fully  developed,  W!  s  that  of  Hansen.  But,  were  this 
method  adopted,  all  that  had  previously  been  done  would  have  been,  useless,  even 
for  the  purpose  of  comparison  and  verification,  owing  to  the  expression  of  the  co- 
ordinates in  terms  of  a  disturbed  mean  anomaly.  It  appeared  to  the  author  that, 
although  this  form  of  theory  led  to  expressions  having  fewer  terms  than  the  other, 
it  was  not  without  its  relative  disadvantages.  Other  considerations  being  equal,  he 
conceived  that  astronomers  generally  would  gr:^atly  prefer  to  see  the  perturbations 
expressed  directly  in  terms  of  the  time,  owin^'  to  the  ease  with  which  the  results 
of  different  investigators  could  then  be  compared,  and  with  which  corrections  to 
the  Hieory  may  be  introduced. 

Under  these  circumstances  the  method  deicribed  in  the  first  chapter  of  the 
present  paper  was  worked  out.  The  question  how  much  it  contains  that  is  essen- 
tially new  is  one  that  the  author  has  never  clos.'ly  examined:  it  is,  however,  certain 

(iii) 


iv 


PREFACE. 


that  the  mode  of  con8iilering  the  subject  is  well  known,  being  that  employed  by 
La  Place,  Ilerschcl,  De  rontecoulnnt,  Encke,  and  perhaps  others.  The  method 
of  forming  the  required  derivations  of  the  perturbative  function  from  the  analytical 
development  of  that  quantity,  he  has  not  seen  elsewhere. 

With  these  improved  elements  and  methods  the  work  was  recommenced  in  1868. 
The  curlier  investigations  being  merely  provisional,  it  has  not  been  deemed  neces- 
sary to  present  them  in  the  present  work.  Some  of  the  results,  corrected  for 
errors  of  the  older  elements,  are,  however,  given  for  the  purpose  of  comparison. 

Although  this  investigation  has  absorbed  the  greater  part  of  the  author's  leisure 
for  more  than  live  years,  it  is  only  through  the  aid  of  the  Smithsonian  Institution 
and  Nautical  Almanac  that  he  has  been  enabled  to  bring  it  to  a  conclusion  witliin 
that  time.  At  an  early  stage  of  the  work  Professor  Henry  responded  favorably  to 
a  request  for  aid  by  the  employment  of  computers;  it  was,  however,  not  found 
practicable  to  use  such  aid  until  the  perturbations  had  been  completed,  and  the 
provisional  theory  concluded.  Then,  the  comparison  of  theory  and  observation, 
and  the  construction  of  the  tables,  involved  a  large  amount  of  mechanical  compu- 
tation, and  on  this  part  of  the  work  a  number  of  persons  have  been  employed  by 
the  Institution  at  various  times,  among  whom  may  be  mentioned  Prof(;ssor  F.  W. 
Uardwell,  of  the  University  of  Kansas,  and  Dr.  C.  L.  F.  Kumpf,  late  of  the  Ob- 
servatory of  Leiden.  Every  part  of  the  work  has,  however,  been  done  under  the 
author's  immediate  direction,  and,  as  nearly  as  possible,  in  the  same  way  as  if  he 
had  done  it  himself,  a  result  which,  in  one  or  two  cases,  has  been  attained  only 
by  the  expenditure  of  an  amount  of  labor  approximating  that  saved  by  the  employ- 
ment of  the  computer. 

In  presenting  the  steps  of  the  investigation,  the  end  has  been  kept  constantly  in 
view  to  render  as  easy  as  possible  the  detection  and  correction  of  any  error,  or  the 
introduction  of  any  alteration  in  the  elements  or  other  data.  It  is,  of  course, 
impossible  to  present  the  steps  of  the  computation  with  any  approach  to  fulness 
without  far  transcending  the  limits  of  the  printed  work :  The  results  given  are, 
therefore,  those  which  it  was  supposed  would  be  most  useful  to  the  futme  investi- 
gator of  the  same  subject.  There  is  reason  to  believe  that  the  original  computa- 
tions will  ultimately  become  the  property  of  the  National  Academy  of  Sciences,  so 
that  they  may  always  be  referred  to  for  the  clearing  up  of  any  difficulty  in  the 
printed  text. 

The  author's  acknowledgments  arc  due  to  Professor  J.  II.  C.  Coffin,  Superin- 
tendent of  the  Nautical  Almanac,  and  Mr.  E.  J.  Loomis,  of  the  Nautical  Almanac 
Office,  for  reading  the  proof  sheets  of  the  last  twelve  tables  during  the  absence  of 
the  former  abroad. 


k 


W'AsnisaTON,  July  31,  1813. 


TABLE  OF  CONTENTS. 


Introduction   . 


PAOI 

1 


CHAPTER  I. 

METHOD  OP  DETERMININfUTnE  rERTrRBATIONS  OP  THE  T.ONOlTrDE,  RADIUS  VECTOR,  AND 
LATITUDE  OF  A  1'I.ANET  BY  DIRECT  INTEORATIO.V. 

Notatfon  and  general  differential  formulB)         ..... 

Formation  of  the  re'.iiired  derivatives  of  the  pcrturbntive  ftmction     . 
Correction  of  these  derivatives  for  terms  of  the  second  order  . 
Integration  formula!  for  perturbations  of  radius  vector 
Development  of  functions  of  rectangular  co-ordinates  .... 

Integration  of  perturbations  of  radius  vector  ..... 

Formnho  for  perturbations  of  longitude  to  terras  of  the  second  order 
Motion  of  the  orbital  planes     ....... 

Perturbations  of  the  second  order  depending  on  the  motion  of  the  orbital  planes 
Reduction  of  the  longitude  to  the  ecliptic        ..... 

E.\pressions  for  the  latitude      ....... 


R 
10 
12 

14 

n 

23 
24 
25 
27 
29 


CHAPTER  II. 

APPLICATION  OP  THE  PRECEDINO  METHOD  TO  THE  COMPUTATION  OP  THE  PERTURlLVTIONa 

OF  URANUS  BY  SATURN. 

Data  of  computation     ..... 
Numerical  expressions  for  R  and  its  derivatives 
Perturbations  of  radius  vector 

Perturbations  of  longitude        .... 
Perturbations  of  latitude  .... 


81 
34 
44 

49 
61 


I 


CHAPTER  III. 

PERTURBATIONS  OP  URANUS  PRODUCED  BY  NEPTUNE  AND  JUPITER. 

Adopted  elements  of  Xcptnno  ..... 
Development  of  li  and  its  derivatives  for  the  action  of  Neptuno 
The  term  of  long  period  between  Xcptune  and  Franus 
Perturbations  of  the  longitude  produced  by  N'cptnno  . 
Perturbations  of  tlio  radius  vector  prndneed  by  Neptune 
Perturbations  of  the  latitude  produced  by  Neptune      . 
Perturbations  produced  by  Jupiter 


53 
64 
55 

58 
fiO 
CI 
63 


(V) 


yl  TABLE   OF   CONTENTS. 

CnAI'TEIl  IV. 

TEBMB  OF  THE  SECOND  OKDEIl  DUE  TU  THE  AOTIOM  OF  BATDRN. 

rrrlimiiiary  iiiTCHtipation  of  the  orbit  of  So,tiirn  .... 

PcrturbatiniiH  of  Saturn  mid  Urnnus     ...... 

Formation  of  llip  oxprossioiiH  for  tlio  terms  of  tlic  second  order 

I'erlurbatioMH  dopendins  on  the  s(|iiarc  of  the  mass  of  .Saturn 

Perlurbatious  depending  on  the  product  uf  the  muHava  of  Jupiter  and  Saturn 

Cn AFTER.  V. 

COLLECTION  AND  TKAN8F0UMATI0N  OF  THE  PRECEDINO  PEKTt^IlBATIONB  OF  URANVB. 

Terms  Independent  of  the  position  of  the  di-sturbiiig  planet 
Secular  Tariation.s  ..... 

Au.xiliury  expressions  on  which  the  perturbations  depend 
Kcduced  expressions  for  the  latitude  of  Uranus 
Positions  of  Uranus  resulting  from  the  preceding  theory 
Elements  III  of  Uranus  .... 

CHAPTER  VI. 

REDCCTIOS  OF  THE  0BSEBVATI0N8  OF  URANl'9,  AND  THEIR  COMPARISON  WITH  THE 

PHECEDINO  THEORY. 

Rcdnction  of  the  ancient  observations  . 

Tlieir  coniporison  with  the  provisional  theory  . 

Discussion  of  the  modern  observations 

Reduction  of  the  results  to  a  uniform  system    . 

Adopted  positions  of  fundamental  stars 

Discussion  of  corrections  to  reduce  the  diiferent  observations 

Table  of  these  corrections         .... 

Results  of  the  observations  from  1181  to  1830 

Observations  from  1830  to  I8t2 

Table  to  convert  errors  of  right  ascension  and  declination  of 

and  latitude  .... 

Tabular  summary  of  results  of  observations,  1830  to  1872 
Corrections  to  be  applied  to  (i  -^  positioi.a  of  Uranus  in  the  Berlin  Jahrbuch  and  the  Nautical 

Almanac  to  reduce  them  to  positions  from  the  provisional  theory 


CHAPTER.  VII. 

FORMATION  AND  SOLUTION  OF  THE  EQUATIONS  OF  CONDITION  RESULTING  FROM  THE 
PRECEDINO  COMPARISONS. 

Expressions  of  the  observed  corrections  to  the  longitudes  of  the  provisional  theory  in  terms 
of  the  corrections  to  the  heliocentric  co-ordinntes  ..... 

Expressions  of  the  same  quantities  in  terms  of  the  corrections  to  the  elements  of  Uranus  and 
the  mass  of  Neptune  ...... 

Table  to  express  errors  of  heliocentric  co-ordinntes  as  errors  of  elements 

Discussions  and  .solutions  of  the  equations  thns  formed 

Concluded  corrections  to  the  elements  of  longitude 

Corrections  to  the  inclination  and  node  of  Uranus 


to  a  homogeneous  system 


Uranus  into  errors  of  longitude 


that 
V5 
68 
fi9 
7G 
17 


79 
80 
81 
93 
98 
99 


ior> 

110 
111 
111 
113 
115 
120 
122 
126 

127 
131 

151 


158 

101 
162 
105 
173 
173 


TABLK   OP   CONTEiNTS. 


CTIAPTEll  Vlir. 

COMPLETION  AND  ABBANOEMENT  OF  THE  TIIEORV  TO  FIT  IT  FOB  PERMANENT  U8E. 

Correction  of  llio  covfllcients  of  tlio  loiiff  inequality  bctwcon  UnimiH  uikI  N'c|ituue  for  tlio 

ternia  of  tiie  second  order  ..... 
Concluded  elements,  or  cloiuenta  IV  of  Uranus 
Lon((-pcriod  ond  secular  pcrturbationfl  of  tlio  ulcmoiits 
Table  of  these  perturbations  from  A.D.  1000  until  A. I).  2200 
Mean  elements  of  Uranus         ..... 
Expressions  for  tbo  concluded  theory  of  Uranus 

CIlAl'TEll  IX. 

OENEBAL  TABLES  OF  UBANU8. 

Enumeration  of  the  qnantitics  contained  in  the  several  tables 
Precepts  for  the  use  of  the  tables         .... 

K.xainplea  of  the  use  of  the  tables        .... 

Tables  of  Uranus  ...... 

Subsidiary  tables  ...... 


vU 


PAOI 

178 
181 
183 

184 
IS4 
186 


190 
195 

198 
20« 
279 


ERRATA. 


Pages  100  to  105.  In  computing  the  latitude  from  the  provisional  theory  the  values  of  the 
secular  terms  of  iij  and  tk  ou  page  97  have  been  iatcruhaugcd.  The  provisioual  latitude,  therefore, 
requires  the  correction 

— 0"..')3  r  Bin  V  +  0".63  T  cos  v 
where 

\—g+  12°  45'  -|-2<;  sin  g. 

This  correction  is  not  applied  in  the  subsequent  investigation.  Its  circot  would  have  boon  to 
change  the  value  of  b  deduced  on  p.  170  l)y  probably  0".2  or  0".3.  Tlic  effect  on  the  other  tlcuienta 
of  latitude  would  have  been  much  smaller,  and  therefore  unimportant. 

Page  122,  line  1,5.     Add  :  the  corrections  in  the  si.xth  column  being  omittoil. 

Pago  151.  Add  foot-note:  In  forming  these  comparisons  the  corri-'ctions  to  the  heliocentric 
longitude  in  the  si.xth  column  of  the  provisional  cphemeris,  pages  100  lo  105,  are  not  applied. 

Page  159.  Equation  7.  In  this  equation  the  coelTicienta  of  ix  and  ip  have  been  multiplied  by  ^, 
instead  of  f ,  the  factor  of  il.  The  effect  of  this  error  enters  into  all  the  subsequent  results,  but  in 
the  compari.sons  of  theory  and  observation  it  is  corrected. 

Page  184.  The  element  hero  represented  by  >  (kappa)  is  the  same  which,  in  the  preceding 
chapters,  has  by  mistake  been  represented  by  k,  and  which  is  defined  on  p.  24.  The  k  of  Chapter 
VII 1  is,  therefore,  not  the  same  with  that  of  preceding  chapters. 


ON  TIIK  ORBIT  OF  URANUS. 


INTRODUCTION. 


The  connection  of  (lie  [jlunot  Uranus  witli  the  most  brilliunt  astronomicnl 
acliicvenjcnt  of  tlic  century  lends  u  peculiar  interest  to  its  theory.  The  researches 
of  Adams  and  Le  Vcrrier  showed  that  the  observed  motions  of  that  planet  were 
represented,  at  least  approximately,  by  the  action  of  i  theoretical  planet  havinj^ 
the  lonj^itude  of  Neptune.  I'eirce  showed  that  the  ih  tion  of  Neptune  itself 
aecoiuited  for  tiiese  motions  within  the  limits  of  possible  error  of  the  observations 
nsi'd  by  Lc  Verrier.  It  remains  to  be  seen  wlicth'-'-  the  n^'reement  between  theory 
and  observation  still  subsists  wiien  the  comparati'  ';  few  observations  used  by  those 
investi.;  itors  are  reduced  with  tlie  more  relined  data  now  at  our  disposal,  and  when 
the  great  mass  of  additional  observations  made  both  befun;  and  since  the  date  of 
Lc  \'errier's  researches  are  included. 

The  circumstances  connected  with  the  discovery  of  Neptune  have  been  so 
exhaustively  recounted  by  a  number  of  authors  that  it  would  be  difficult  to  add 
anything  not  already  familiar  to  astronomers  without  transcending  our  present 
limits.  I  shall  therefore  confine  myself  to  such  nn  account  of  previous  researches 
on  the  theory  of  Uranus  as  may  give  an  idea  of  their  nature  and  extent,  and  facili- 
tate their  comparison  with  the  methods  and  results  of  the  present  investigation. 

The  perturbations  used  by  Bouvard  in  his  tables  are  those  of  the  ^lecanique 
Celeste.  Although  not  affected  with  any  striking  error,  the  numerical  methods 
adopted  in  their  computation  are  necessarily  too  rough  to  allow  of  much  interest 
attaching  to  their  comparison  with  the  results  of  the  more  recent  researches. 

It  is  essential  to  a  clear  understanding  of  subsequent  researches  that  we  classify 
the  methods  which  have  been  or  may  be  adopted  in  the  computation  of  the 
general  perturbations  of  the  planets.  This  computation  comprises  two  distinct 
operations:  (1)  the  development  of  the  disturbing  forces,  or  some  quantities  of 
which  these  forces  are  fmictions ;  (2)  the  integration  of  the  equations  of  motion 
inider  the  influence  of  these  forces.  In  each  of  these  operations  three  methods 
have  been  employed. 

In  developing  the  perturbativc  function,  we  have  first  the  purely  analytic  method 
used  by  the  great  geometers  of  the  last  century.  In  this  metliod  this  function  is 
developed  in  powers  of  the  eccentricities  and  mutual  inclination  of  the  orbits  of 
the  two  planets,  and  the  numerical  coefficients  are  found  by  substituting  the  values 
of  the  elements  in  these  expressions.     A  is  only  applicable  when  the  eccentricities 

1         Uaroh.lSTS.  (   ^  ) 


:i\ 


THE   ORBIT   OF   URANUS 


and  mutual  inclination  are  small,  and  has  for  that  reason  fallen,  of  lute,  into  a 
certain  disrepute.  The  extended  tables  publislied  Ly  Le  Verrier'  liave,  however, 
added  so  much  to  its  facility  for  use  that  it  is  not  wholly  unworthy  of  attention. 

At  the  other  extreme  stands  the  purely  mechanical  method,  in  which  spccifil 
values  of  the  disturbing  force  are  computed  for  many  combinations  of  the  mean 
anomalies  of  the  two  phniets,  and  tlic  values  of  the  coefficients  in  the  general 
expression  for  the  force  thence  deduced. 

Between  these  two  stixnds  what  1  conceive  we  may  designate  as  the  Cauchy- 
Hansen  method,  in  which  the  development  is  made  mechanically  with  respect  to 
the  one  ])lanct,  but  the  eccentric  anomaly  of  the  other  is  retained  as  an  undeter- 
mined quantity.  The  germ  of  this  mctliod  is  found  in  several  papers,  by  Cauchy, 
in  the  earlier  volumes  of  tlie  Comiitca  Rendiis  of  the  French  Academy,  which  have 
since  been  combined  into  a  liomogeneous  memoir  by  Puiseux.^  Tlie  object  had  in 
view  by  these  authors  is  only  the  computation  of  inequalities  of  long  period.  But 
Hansen  has  taken  up  the  essential  principle  of  the  metliod,  first,  in  his  piize  memoir 
on  the  perturbations  of  comets,  crowned  by  the  French  Academy  of  Sciences,  about 
1848,  and  afterwards  in  his  "  Auseinaiiderscfzuiiff  ei'ner  zwcckmiimgcn  ilethode  znr 
JJcrecIinung  tier  Storiiugcn  (hr  klcinen  Planetcn"^  and  applied  it  to  the  general 
development  of  perturbations 

Among  the  three  metliods  of  integration,  the  first  in  point  of  analytical  elegance 
and  generality,  but  the  last  in  order  of  convenience  in  use,  is  that  of  the  variation 
of  elements,  a  method  with  which  the  name  of  La  Grange  is  inseparably  associated. 

In  the  second  the  direct  integration  of  the  differential  equations  wliich  express 
the  perturbations  of  longitude,  latitude,  and  radius  vector  is  effected  by  special 
devices. 

In  the  first  of  these  methods  the  problem  is  presented  in  this  form :  The  equations 
of  motion  being  completely  integrated  for  the  action  of  the  principal  forces  only, 
how  must  the  arbitrary  constants  of  integration  vary  in  order  that  the  same  expres- 
sions may  represent  the  motion  of  the  planet  under  the  influence  of  the  disturbing 
forces?  In  the  second  method,  the  same  thing  being  presupposed,  the  question  is, 
what  expressions  must  be  added  to  the  integrals  of  undisturbed  motion  in  order 
that  the  sum  may  represent  the  integrals  of  the  disturbed  motion  X 

The  third  is  Hansen's  method,  in  which  the  co-ordinates  are  partly  expressed  in 
terms  of  a  certain  function  of  the  time  known  as  th?  disturbed  mean  anomaly, 
determined  by  the  condition  that  the  true  longitude  in  the  disturbed  orbit  shall  be 
the  same  function  of  the  disturbed  time  that  the  longitude  in  tlie  elliptic  orbit  is 
of  the  simple  time. 

Although  the  last  two  metliods  have  a  great  advantage  over  the  first  in  the  com- 
putation of  the  periodic  perturbations,  I  conceive  the  first  to  be  best  adapted  to 
the  computation  of  the  secular  variations,  and  perhaps,  of  terms  of  very  long  period 
in  the  mean  longitude  and  the  elements  of  the  orbit. 


•*■ 


•  Annales  de  V Ohservatoire  Imperial  de.  Paris.     Tome  I. 

'  AiinnleK  de  roharrvaloire  Jmpi'rinl  de  Paris.     Tome  A'll. 

•  AbhamUitnrjen  der  KOniglich,  Sdchsischcn  Oescllscha/t  der  Wisscnscha/lcn.    Band  V.  VI,  VII. 


•I* 


THE   ORBIT   OP    URANUS. 


3 


In  his  researches  on  the  motion  of  Uranus,  the  first  thing  done  by  Le  Verricr 
was  to  reconii»ute  the  perturbations  by  Jupiter  and  Saturn.  It  will  sufficiently 
describe  his  method  of  doing  this  to  say  that  in  the  developments  he  used  the 
purely  mechanical  method  for  the  action  of  Saturn,  and  the  algebraic  development 
of  tlie  perturbative  function  for  the  action  of  Jupiter,  while  in  the  integration  he 
used  the  method  of  the  variation  of  elements.  After  completing  the  perturbations 
of  the  first  order  he  made  the  earliest  attempt  at  a  complete  determination  of  those 
of  the  second  order.  Beginning  with  the  terms  of  tliis  order  which  arise  from  the 
secular  variations  of  the  elements,  he  determines  them  by  recomputing  the  terms 
of  the  first  order  for  the  epoch  2300,  and  assuming  that  the  general  term  will  then 
be  given  by  interpolating  between  the  two  terms  thus  found,  supposing  them  to 
increase  uniformly  with  the  time.  This  proceeding  has  the  sanction  of  such  high 
authority  that  it  is  worth  while  to  call  attention  to  its  want  of  rigor.  The  dif- 
ferential coefficient  of  each  element  being  given  in  the  form 

(la       J         J, 
,   =  k  cos  Ot. 
(It 

k  being  a  function  of  the  elements,  the  perturbation  of  the  first  order  will  be 

(a  =  ,  sua  lit. 
b 

When  we  take  into  account  (he  variation  of  /•,  and  suppose  it  of  the  form  h^  -}-  Ic't, 
the  process  is  equivalent  to  supposing  that  in  this  case 

la  = '  —  sni  bty 

0 

whereas  it  really  contains  the  additional  term, 

J  j  cos  bt, 

which  appears  to  be  neglected  in  tlic  process  in  question.  It  will  be  seen  that  the 
neglected  coefficient  is  equal  to  the  secular  variation  of  the  term  during  the  time 
that  its  argument  requires  to  increase  by  an  amount  equal  to  the  unit  radius.  It 
is  tlierefore  the  more  important  the  longer  the  period  of  the  inequality. 

To  obtain  the  periodic  terms  of  tlie  second  order  Le  Verrier  begins  by  determin- 
ing the  ten  principal  terms  of  the  perturbations  of  tiic  elements  of  Saturn  produced 
by  Jupiter.  Next  he  takes  up  the  terms  in  the  mean  longitude  of  Uranus  which 
depend  on  the  square  of  the  mass  of  Saturn.    The  only  sensible  terms  he  finds  are 

—  T'.n  sin  {^—  30  — 0".35  cos  (^'  —  S^) 

+  0".43  sin  {i,"  —  ^'  +  40  —  0".2l  cos  (^'  —  4^'  +  40, 

^,  <^',  and  ^"  being  the  mean  anomalies  of  Uranus,  Saturn,  and  Jupiter,  respectively. 
The  terms  depending  on  the  product  of  the  masses  of  Jtq)iter  and  Saturn  are  then 
taken  up.  lifteen  arguments  are  found  the  coefficients  of  wliich  vary  from  a 
small  fraction  of  a  second  to  one  or  two  seconds,  while  a  single  one  of  long  period 
amounts  to  32". 

When  the  method  of  variation  of  elements  is  used,  it  is  necessary  not  only  to 
determine  these  variations  to  quantities  of  tlie  second  order,  but,  in  the  tiansforma- 


THE   ORBIT   OP   URANUS. 


tion  of  the  perturbations  of  the  elements  into  perturbations  of  the  co-ortlinates,  to 
carry  tliis  transformation  to  terms  of  the  second  order  also.  This  Le  Verrier  avoids 
by  showing  that  the  terms  of  the  lowest  order  with  respect  to  the  eccentricities 
thus  introduced  are  destroyed  by  certain  terms  in  the  perturbations  of  the  elements, 
so  that  it  is  only  necessary  to  omit  both  classes  of  terms.  These  terms  are  of  that 
fictitious  class  which  disappear  of  themselves  by  a  simple  change  of  elements. 
When,  instead  of  the  eccentricity  and  longitude  of  the  perihelion,  we  take  h  and  /■•, 
which  represent  the  products  of  the  eccentricity  into  the  sine  and  cosine  of  this 
longitude  respectively,  these  terms  disappear  of  themselves  both  from  the  perturba- 
tions of  the  elements  and  of  the  co-ordinates.  It  is  not  likely  that  any  of  the 
neglected  terms  of  tliis  class  exceed  0".l. 

As  soon  as  the  elements  of  Neptune  were  known,  the  nature  of  its  general  action 
on  Uranus  became  of  interest.  This  subject  was  taken  up  by  Prof.  Peirce,  whose 
results  are  found  in  the  Proceedings  of  the  American  Acatlemy  of  Arts  and  Sciences, 
Vol.  T,  pp.  334-337.  This  paper  is  accompanied  with  a  comparison  of  his  theory 
of  Uranus  with  observations,  to  wliich  simihir  comparisons  of  the  theories  of  Adams 
and  Le  Verrier  are  added.  Tliis  comparative  exhibit  is  of  sufficient  interest  to  be 
given  here.  The  numbers  given  arc  probably  excesses  of  computed  over  observed 
longitudes. 


Residual  Differknceb  between  the  Theoreticai,  anp  Observed  Longitudes  of  Uranus,    | 

riio.M  the  Tueouies  of  1'eirce,  Le  A'ekrieb,  a.nd  Adams. 

From    he  Ver- 

From    Le   Ver- 

From    Adams's 

From  I'eiroe's  theory  of  Neptune 

adopting  for  its 

rier'a  test  or- 

rier's  original 

original     the- 

mass 

ory   with    his 

from   tlie  mo- 

hi!) best  orbit 

se(!0iid    hypo- 

That  of  Strnve 

Tliat     deduced 

That    deduced 

dern  ob^^erva- 

of    hypotheti- 

tlietical planet 

from  his  own 

by  Peirce  from 

by  Peirce  from 

Year. 

tions   without 

cal  planet,  of 

of  which  mass 

observations 

Bond's  &  Las- 

Bond's  obser- 

any    external 

which         the 

is 

of  the  satellite 

sel'a  observa- 

vations of  Las- 

planet. 

mass  IS 

tions  combined 

sel's  satellite 

jjVi- 

zi^-^- 

Tr'tT^' 

riHi- 

•n\n- 

ft 

// 

// 

// 

// 

// 

1000 

+  289.0 

—  19.9 

+  50.0 

—  124.7 

+  13.0 

-f  0.8 

1715 

-1-  279  0 

+    5.5 

—    0.6 

—    99.6 

4-  lO.O 

+  8,7 

1750 

+  230  9 

—    40 

—    4.0 

—  102.4 

—  12.7 

+  4,0 

1709 

-Y  123  3 

+    3.7 

+    1.8 

—    67.0 

—  10.0 

—  0,0 

1782 

+    20  5 

+    23 

0.0 

—    18.3 

—    5.6 

—  3.0 

1787 

+      2,0 

—    12 

—    0,2 

—      4.7 

—    12 

—  0,5 

1792 

—      7.8 

+    0  3 

—    1.1 

+      1.0 

-f    0  5 

+  0,3 

1797 

—      6.7 

—    1.0 

—    0,;') 

-f      3  3 

-f    0.8 

+  0,3 

1803 

—      34 

+    0.8 

+    Ifi 

+      3.2 

4-   1.2 

+  0,8 

1808 

+      3.8 

+    0.8 

0.0 

—      1.3 

—    0.6 

—  0,4 

1813 

+      4.5 

—    09 

—    1.0 

—      2.3 

+    M 

—  0,3 

1819 

+      3.8 

+    0.4 

—    22 

+      0.9 

+    0.7 

+  1.0 

1824 

—      7.0 

—    54 

+    1.1 

—      1  6 

—    1.9 

—  2.0 

1829 

—      7.8 

—    2.2 

+    2.0 

+      25 

+    1.3 

+  0.8 

1835 

—      45 

—    0,8 

—    1.2 

-f      3.9 

+    24 

+  2.0 

1840 

+      07 

+    22 

+    1.3 

-      1.3 

—    1,3 

—  1.1 

1845 

+      6.5 

—    0,3 

—      2.8 

—    1  2 

—  0.9 

■4 


i 


In  this  paper  Professor  Peirce  presents  the  results  of  a  complete  computation  of 
tlie  general  perturbations  of  Uranus  by  Neptune  in  longitude  and  radius  vector, 


THE   ORBIT   OF   URANUS. 


I  r 


but  witliout  any  details  v/hatcvcr  of  the  investigation,  or  any  statement  of  the 
methods  employed.  Tlie  minuteness  of  the  residuals  in  the  last  column  of  the 
preceding  table  shows  that  employing  these  perturbations  by  Neptune,  and  those 
of  Le  Verrier  by  Jupiter  and  Saturn,  we  had  a  theory  of  Uranus  from  which  quite 
accurate  tables  might  have  been  constructed.  But  this  never  seems  to  have  been 
done.  The  cphemeris  of  Uranus  in  the  American  Nautical  Almanac  was  intended 
to  be  founded  on  this  theory,  but  the  proper  definitive  elements  do  not  seem  to 
have  been  adopted  in  the  computations,  as  the  cphemeris  does  not  correspond  with 
the  theory. 

Although  twenty-five  years  have  elapsed  since  the  epoch  of  these  researches,  I 
am  not  aware  of  any  published  work  of  importance  on  the  theory  of  Uranus  during 
the  interval.  Mr.  T.  H.  Safford  has,  however,  made  a  very  extended  investigation 
of  the  subject,  but  has  published  nothing  more  than  a  brief  general  description  of 
his  work,  which  may  be  ^ound  in  the  Monthly  Notices  of  the  Royal  Astronomical 
Society,  Vol.  22.  Like  Professor  Teirce,  he  took  I^e  Vcrrier's  perturbations  by 
Jupiter  and  Saturn,  but,  instead  of  using  general  perturbations  by  Neptune,  he 
computed  the  effect  of  the  action  of  this  planet  by  mechanical  quadratures  for  the 
whole  period  of  the  observations  of  Uranus,  and  thus  corrected  the  elements  and 
the  mass  of  Neptune  from  modern  observations  alone.  The  mass  in  question 
deduced  was 

1 
20039 
Mr.  Safford  does  not  give  the  representation  of  the  modern  observations,  but  pre- 
sents the  following  comparison  of  the  ancient  ones,  alongside  wliicli  wo  place  for 
comparison  the  corresponding  numbers  of  Pcirce's  theory  and  those  of  the  present 


investigation. 

Excess  op 

Observe 

Date. 

No.  of  obs. 

Safford. 

ifino 

1 

+  5".0 

,        1-15 

3 

—  4.2 

ltr>o 

2 

—  1  2 

It  53 

1 

_  0.2 

1"5G 

1 

—  0.0 

17G4 

1 

+  0.4 

nco 

8 

+  4.5 

Peirce. 

Newcomb 

—  0".8 

-u" 

—  8.T 

—    8 

—  4.0 


+  CO 


+    2.0 


—  1.4 


6 


THE   OllBlT   OF    UKANUS. 


CHAPTER  I. 

METHOD  OP  DETERMINING   THE   I'EUTURIJATIONS  OF  LONGITUDE,  RADIUS 
VECTOR,  AND  LATITUDE  OF  A  I'LANET  BY  DIRECT  INTEGRATION. 

Let  us  conceive  a  plane  determined  by  the  condition  tluit  it  shall  pass  through 
the  sun  and  contain  the  tangent  to  the'  orbit  of  a  planet  at  any  moment.  If  the 
planet  were  acted  on  by  the  sun  alone,  the  position  of  this  plane  would  be  invariable, 
but,  under  the  influence  of  the  disturbing  forces  of  the  other  planets,  it  is  subject, 
at  each  instant,  to  a  motion  of  rotation  around  the  radius  vector  of  tlie  planet.  We 
may  regard  this  as  the  instantaneous  plane  of  tlic  plaiu>t's  orbit.  Tlie  disturbing 
and  the  disturbed  planet  will  each  have  its  own  instantaneous  plane. 

Let  us  now  put : — 

V,   the  longitiule  of  a  planet  counted  from  a  determinate  point  in  the  instantaneous 
plane  of  its  orbit. 

V,  its  distance  from  the  node  of  intersection  of  its  own  orbit  with  that  of  another 
planet, 
the  mutual  inclination  of  the  two  orbits, 
sin  I  y. 

the  radius  vector  of  the  planet, 
its  logarithm. 

fi,  the  attractive  force  of  the  stui  upon  unit  of  matter  at  unit  distance. 
a,  the  mean  distance  corresponding  to  the  observed  mean  motion  of  the  planet, 
determined  by  tlie  condition 

»r 
m  and  n  being  as  usual  the  mass  and  mean  motion. 
a„,  the  value  of  a  corrected  for  the  constants  introduced  by  the  perturbations,  so 
that,  as  in  the  elliptic  motion,  we  have 

p  =  logrt+/(/,  p,  ct), 
we  shall  have  in  the  disturbed  motion 

p  =  log  CTo  +/(^  e,  ct)  -|-  periodic  terms  only. 

«!,  the  mean  distance  of  an  outer  planet,  whetlier  it  be  a  disturbing  or  disturbed 

planet. 
»,  the  logarithm  of  a. 

a,  the  ratio  of  two  mean  distances,  taken  less  than  tmity. 
li,  the  perturbative  function. 


o, 
r, 

P 


%« 


Mt«9 


THE   ORBIT   OF   UIl ANUS.  J 

h,  the  coefficient  of  any  term  of  -i-  R,  so  that  we  have 

B  =  Z COS  N 

«1 

m'  being  here  the  mass  of  the  disturbing  planet. 
X,  the  mean  distance  of  the  planet  from  the  node,  or  the  mean  vahic  of  v. 
w,  tlie  distance  of  the  perihelion  from  the  node. 
g,  the  mean  anomaly. 

I,    the  mean  longitude,  or  the  mean  value  of  v. 
4/,  the  angle  of  eccentricity  so  that  c  =  sin  t^^. 
r„  the  radius  of  the  planet  in  the  undisturbed  ellipse, 
r,,  the  quotient  of  r„  divided  by  the  mean  distance,  wliich  is  a  function  of  the 

eccentricity  and  mean  anomaly  only. 
T,  the  time  after  the  epoch  1M50,  Jan,  0,  Greenwich  mean  noon,  counted  in  Julian 

centuries. 

V,   the  integrating  factors  of  the  periodic  terms,  or  the  ratio      ,  ^  being  the  change 

of  the  angle  in  iniit  of  time. 
«,  the  eccentric  anomaly,  and,  in  tlie  tables,  the  argument  of  latitude. 
We  have  for  the  value  oi  li 


Mt»9 


It 


m 


m  r 


(cos  v  cos  v'-)-sin  V  sin  v' cos  y) 


V 1^ — 2)-r'(cosvcosv'-|-sin  v  sin  v' cos  y)-\-r'''       i 
or,  if  we  suppose  r  replaced  by  its  value  in  p,  namely 

we  shall  have 

With  this  value  of  R  it  is  well  known  that  the  differential  equations  for  the  longi- 
tude and  radius  vector  of  a  planet  arc 


dh- 


dr^       "  (!+»')_      ^J^ 


de 

d('  ^ 

r 

fPv 

de 

+  2, 

dr  dv 
dt  dt  ~~ 

(^li 

=  F 


cJp' 


(1) 


If  we  multiply  the  first  of  those  equations  by  2  '  P  and  the  second  by  2  J^  and 


dt 


dl 


add  them  together,  putting,  for  brevity, 


f*^^  (/p   1^  r/?  dv jy  „ 

cJp  f/<+  ov  dt  -^^''^- 


(2) 


and  then  integrate,  we  shall  have 


';:+y';:-'^'-^'+'"'='>,(c+fir.n,o 


d( 


8 


THE   ORBIT   OF   URANUS. 


^~d^^ ~=t^{2c+2fiy,m+f)     c3) 

Lot  us  now  represent  by  ,,  that  elliptic  value  of  .  which  satfsfies  the  equation 
Subtracting  this  equation  from  the  last  we  have 

^^t::zt^:js::^:::tz:'h'''':  -'  '-'^  -''-  ^^^  ^^-^o- 

the  logarithm  of  the  radh  ^  ."  he  .t    ,7  "  ""  'T'f  ''^'^""^'''^'^  "^  ^'"^'"S 

for  the  latter  its  value  '""'''''  '"^'^'''^  °^  '^  "^'-'If'  ^^c  substitute 


for  the  latter  its  value 

and  put 

Then 


^P  =  P  — po. 


^('•'-0  =  »v^p-f,-„^y^etc. 

Substituting  these  valiip<!  in  fl,«  .,1, 

tern,  of  ,|,e°  second  orfTr    „d  t"™™:?  h ''"''l  '""^'"K, ""'  ''"^P-onl  „„,y  „ 

J*  /2S\  t  *     ^         I  />        ^    Hilt.. 

"  dt^  +  &  ( V5p)  =  fc  ( 2Sjyjidt  +  ^\_<^  (ro'V)  ,  f^  ¥      rx. 
•  ^    ^  ^  dp)  dC'        I- -9,.  '       (*) 

an  equation  which  gives  the  perturbations  of  radius  vector 

we  siLi  b4in  by  ^:::!:ir;^i:^^:::zz:t:7  "^  ^"'' '-'-' ''-  '"--^ 

of  5v,  K  ^p,  ap',  ^y,  and  that  from  the  e  we  wi  h    0  "  ^r""''''? ^  '"  *'"  "•'"'''^ 
We  must  first  carrv  this  innrnv,-..  .      .     .   ^    '  *"  ""  '"''"""'^  approximation, 

ber  of  (4).  To  effLuhi^rr;  7  '",  ''V'  '""^'""^  "^  ^  '"  ^'^  ^^""^  "-- 
of  the  elements  and  ,  time  w  mtT  "'  '™'"  '"  ^--'"P--^*  of  ^  i"  terms 
to  the  quantities  which  or 'iJo  ^^  ^Zu^'^'^t  't^'^''  '''''  ^^^"^^^ 
and  r,  is,  in  its  developed  form,  a  futtbn  of^T  '  '>  '""?'""  "5  ^''Z '  P'  P'' 
development  being  effected  bv  s nbsfih  f  /  u  L"'  "'  ^'  "'  "'  "'  ""'*  ^  '^'^ 
values  in  terms  of  the  second  '"''''''*'"?  ^"'^  ^''^  «-*  set  of  quantities  their 
01  me  second.     Ihe  substitution  is  as  follows: 

v  =  ;i  +  /},, 

^-^'+^y,  (5) 

P  =  »  +  «/'5' , 
P'  =  »'+  <?'^', 


THE  ORBIT  OF  URANUS.  9 

Fg  being  the  equation  of  the  centre,  and  ^g  the  part  of  p  depending  on  the  eccen- 
tricity in  the  elliptic  motion.  It  follows  that  if  wc  express  the  developed  expression 
for  iJ  as  a  function  of  X,  X,  g,  ^,  »,  x/,  which  we  may  do  by  putting 

u  =  /I  —  g,  (J  =  W  —  (f; 
a  =  c',         a!  =  c"'; 

we  shall  have  by  successive  differentiation 

~dX*  ~  5v*  'd-^  ~  6\* 
6R  _  6R  «?p  _  6R 
dTO~~    dp    dv~   6p 

d-a"  ~  dp"'  d-o  ~  5p» 
etc.  etc.  etc. 


(6) 


and  in  general 


^m  +  n  +  m'  +  n-^  ^m+n+m'+n'^ 


5X"  5»"  ^A,""'  5»"''      Sv™  df  oiv'^'^p" 


Thus,  by  expressing  the  developed  R  in  the  above  form,  we  may  find  the  derivative 
of  any  order  with  respect  to  v,  v',  p  and  p',  by  taking  the  corresponding  derivative 
with  respect  to  X,  7!,  »  and  »'. 

The  developed  R  is  usually  expressed  in  the  form 

12  =  2  -"*'*  cos  {i'X  +  a  +y'(j'  +io)) 

a,  being  the  mean  distance  of  the  outer  planet,  whether  disturbing  or  disturbed, 
and  h  a  function  of  e,  e",  a,  and  y.  Substituting  for  o  its  value  in  g,  this  equation 
will  become 

72  =  2  ^^cos  ((»'+/)  7:-^r(i+jn-jy-jg\ 

Putting  for  brevitv 
the  formulae  (6)  give 

^v  a, 

|^  =  -2^(^+yVcosiyr 


0) 


and  in  general 

"dv^W"'  dp"  dp" 

a        Htroh,  1873. 


^  "  +  •»'_ 


JV 


hr 


10 


THE   ORBIT   OP  URANUS. 


The  formation  of  the  derivatives  in  the  second  member  of  this  equation  demands 
attention.  In  the  analytic  development  of  the  perturbativc  function  each  value  of 
h  is  composed  of  a  scries  of  terms  each  of  the  form 

Ex  A, 

E  being  a  function  of  ♦he  eccentricities  and  mutual  inclination,  and  A  a  function 
of  a  of  the  form 

(0)a'-.6^.  +  (l)a'^'^  +  (2)a-^^^'  +  etc.  +  a^rjg',       (8) 

(0),  (1),  etc.,  being  numerical  coefficients  connected  with  the  coefficients  F"  tabu- 
lated by  Le  Verrier,  in  Tome  I  of  his  Annalcs  de  I'Obscrvatoire,  by  the  relation 

(">=i.J".-.v 

and  J','>  being,  as  usual,  the  coefficient  of  cos  i^  in  the  development  of 

(1— 2acos<j>-fa=)-' 
in  multiples  of  cos  ^,  and  »  —  1  the  sum  of  the  exponents  of  the  eccentricities  in  E. 
It  would  have  been  much  more  convenient  if  in  effecting  this  development  the 
derivatives  of  fe'j'  had  been  taken  with  respect  to  »  instead  of  a.     In  fact  the 

derivative    ^  ^  when  expressed  in  terms  of  the  derivatives  with  respect  to  »  is  of 

da  '^ 

the  form 


«"--•=".--.:-+«. -^„;-  + etc. +  ».,.-^ 


<9a" 


6to 


Therefore,  when  expressed  in  terms  of  the  derivatives  with  respect  to  »,  A  will 
be  of  the  form 

a'-' (  (0)' t<i' +  (ly  ^  +  (2)' 1^ -f  etc. ), 

dA    6^A 
from  which  the  derivatives  -        -     -,  etc.,  may  be  foimd  with  great  facility. 

As  in  the  actual  developments  of  R  which  we  possess,  the  values  of  A  are  given 
in  the  form  (8),  we  must  find  the  expression  for  the  first  two  derivatives  of  its 
several  terms  with  respect  to  »,  which  we  easily  do  by  the  application  of  the  sym- 
bolic formulae 

D\  =  a{Da-{-aLP,). 
Beginning  with  the  case  of  «  =  },  we  have 

'dr^'~'^'6a    ^"-    da?' 


H^^ 


THE   ORBIT  OF   URANUS. 


11 


V       da"/ 


dn 


=  na" 


^x>> 


^a» 


=  71 


oo 


^» 


consequently  wc  have  for  the  derivatives  of  A  from  formuloe  (8) 

6A_ 


=  (0)  a  -^t  +  (1)  (  a  -^  +  a»  -^^,' )  +  (2)  (  2a'  -^^-,i  +  a'  ^j)+  etc. 


5a    ~r  -  ^„,  /  -r  v-y  v-*  "^oT  ^  ""*  ~5a»'  +  "'  ^ 


-)+ 


etc. 


The  derivatives  of  A  being  formed  in  this  way,  those  of  h  are  immediately 
deduced  from  the  equations 

6h    6  A 


oVi 


d^A 


dri'   ~  ^^  dra' 


Wlien  8  is  equal  to  |,  ^  is  of  the  form 

a  I  (0')  i'l'  +  (!)'«  :^'^  +  (2)' a'  ^^''l  +  etc.  I 
I  <9a  ^a"  3 


W 


The  quantity  within  parentheses  is  of  the  same  form  with  A,  in  the  case  of  s  =  |. 
If  we  represent  it  by  A'  we  sliall  liave 

6A  /  dA'  A  SA'     ,     . 


d^A 
dri 


,3   =^  +  2a-^-+a~^^ 

dA       ,  oM' 


A  being  the  same  form  with  A,  tlie  derivatives   ^'"'   and 

5»  (9ir 


will  be  of  the  form 


(9),  substituting  |  for  the  index  |,  and  (0)',  (1)',  etc.,  for  (0),  (1),  etc. 

In  the  case  of  s  =  |  the  derivatives  are  obtained  in  the  same  way,  which  is  too 
simple  to  need  elucidation. 

We  have  now  to  pass  from  the  derivatives  of  h  to  those  of      * ,  the  coefficients 

of  the  perturbative  function.     The  form  of  these  derivatives  will  depend  not  on 
whether  the  planet  is  disturbing  or  disturbed,  but  on  whether  it  is  an  outer  or 


flq^ 


12 


THE   ORBIT   OF   URANUS. 


inner  one.  Let  us  then  suppose  for  the  present,  that  a  and  «  refer  to  the  inner 
planet,  and  put  »,  for  the  logarithm  of  the  mean  distance  of  the  outer  cue.  Wo 
then  have  for  the  derivatives  relatively  to  » 

5- A 

a,       1    d'h 


5»» 


a,  dvf 


n   » 


and  for  the  first  derivative  relatively  to  »,  using  the  symbolic  notation, 


^A 


^  =  1(Z)., -1)7». 


5»j        «i 

The  symbols  in  the  second  member  being  distributive,  vire  have  by  successive 
differentiation 

The  quantity  A  is  a  function  of  o,  the  ratio  of  the  mean  distances  or  oi  C'—'i,  0 
being  the  neperian  base.     Hence 

D.y*  =  —  D,h, 

which  substituted  in  the  last  equation  gives 

h 


6" 


a,       (-1)" 


dK 


(Z>, +  1)»A. 


(10) 


This  formula  gives  for  the  first  two  derivatives 


5»i 
d«A 


Substituting  in  the  general  formula;  (7)  these  expressions  for  the  derivatives 
relatively  to  v  and  »,  we  have  expressions  for  the  derivatives  of  E  relatively  to 
V,  V,  p,  p',  it  being  understood,  however,  that  all  the  quantities  are  expressed  in 
functions  of  the  elements  of  elliptic  motion. 

In  order  to  compute  the  perturbations  of  the  second  order  we  must  carry  E  and 
such  of  its  derivatives  as  enter  into  the  difierential  equations  (1)  to  quantities  of 
the  first  order  with  respect  to  the  perturbations.  Let  us  then  represent  by  Vo,  Vo', 
Po7  p'o»  yo.  t^e  elliptic  values  of  v,  V,  p,  p',  and  y,  which  we  have  assumed  in  the 
Erst  approximation  to  the  perturbations,  and  by  5v,  6v',  etc.,  the  quantities  to  be 


» 


* 


THE  Oil  BIT   OF   URANUS. 


ta 


added  to  v„,  v,,',  etc.,  to  make  the  true  vnluos  of  v,  V,  etc.,  whether  perturbations  or 
corrections  of  the  elements.     \Vc  shall  then  have 


<9v„» 

dm. 


h\  + 


dRo 

d'R„ 


■  W  + 


JR^ 

d'R, 


^  oiVg^v'o         ^    C^VoOlpo 


S'R, 


d'Ro 


^po^Vo        ^  ^pu<5v'«       ^     df\ 


.3p  +  - 
ip  + 
5p  + 


5p'o 
S'Ro 

^V„c)p'o 

^po^p'o 


c'Voo'yo 


Sfodyo 


(11) 


The  value  of  DftR  may  be  found  either  by  cqiuition  (2),  or  by  differentiating  with 
respect  to  the  time  as  introduced  by  the  co-ordinates  of  the  disturbed  planet. 
When  quantities  of  the  first  order  only  are  considered  the  latter  operation  is  very 
simple,  but  it  is  different  when  terms  of  the  second  order  come  in,  because  the  true 
longitude  of  the  planet  is  then  expressed  in  terms  not  only  of  its  own  mean  longi- 
tude, but  also  of  the  mean  longitude  of  all  the  disturbing  planets.  The  result  can 
still  bo  obtained  in  the  same  way  by  separating  all  the  mean  longitudes  introduced 
by  the  co-ordinates  of  the  disturbed  planet  from  those  introduced  by  the  co-ordinates 
of  the  other  until  after  the  differentiation  relatively  to  <'. 

Let  us  now  resume  the  equation  (4),  representing  its  second  member  by  //  Q,  so 
that  it  becomes 


where 


(It  Vq 


(12) 


By  the  operations  already  given  Q  has  become  a  known  function  of  the  time. 

It  is  well  known  that  the  integration  of  (12)  may  be  effected  by  finding  two 
values  of  rj^hp  which  satisfy  this  equation  when  the  second  member  is  neglected, 
or,  in  other  words,  by  finding  two  variables  x  and  y  which  satisfy  the  equations 

<r-x       /^(l  +  vi)  Q 


<^>/    ,    /'(I  +»»)„_  0 


when  the  required  integral  is 


'5p  = 


^ 


Ji/         dx 


ill 


yji 


I  yjxQdt  —  xjyQdtX. 


The  above  differential  equations  are  satisfied  by  the  rectangular  co-ordinates  of  the 
planet  in  its  assumed  elliptic  orbit.  The  position  of  the  axes  of  co-ordinates  being 
arbitrary  we  shall  take  the  line  of  apsides  for  the  axis  of  X,  the  perihelion  being 
on  the  positive  side.     If  we  put 

Co  =  sin  1^, 
we  have 

dy         dx        , :r — - 

■       yj/=A«(i  +  ' 


X 


dt 


dt 


M  (1  +770008  4- 

|cos^= ^„ 


<l^ 


w 


u 


THE   ORBIT   OP   URANUS. 


Lot  us,  for  convonionno,  rpplacc  x  and  y  by  twci  otlier  variables  £  and  yj  connected 
with  them  by  the  equutiuns 

X  =  aC, 

y  =  ari  cos  4'. 

f  and  Yi  are  then  functions  of  the  eccentricity  and  mean  anomaly  only,  and  may  be 
developed  according  to  the  multiples  of  the  latter.  Substituting  the  last  three 
expressions  in  the  preceding  value  of  r^bf  it  becomes 


''»'^p=i+l{''/'^'^'-^^'^'^'M 


If  we  put  r,*  for  the  value  of  r^  when  the  mean  distance  of  the  planet  is  put  equal 
to  unity,  so  tluit  r,,  like  ^  and  yi  contains  only  the  eccentricity  and  mean  anomaly, 
we  shall  have 

hf  =  "^"'^  I  yif^a  Qdt  -  ^fyja  Qdl  |  (13) 

We  mnst  now  express  ^  and  y;  in  terms  of  the  time,  or  of  the  mean  anomaly. 
Putting  for  the  present  n  f  )r  the  eccentric  and  o  for  the  true  anomaly,  we  have, 
by  the  theory  of  the  elliptic  motion. 


from  which  follow 


X  =  >•  cos V ^a  (cos n  —  c), 
y  =  r  sin  v  =  a  cos  ^  sin  m, 

^  =  cos  n  —  e, 

ri  ■=-.  sin  M, 


As  ^  and  >?  are  to  be  expressed  in  the  form 

i=--\^Pi  cos  hj, 

the  finite  integrals  extending  to  all  values  of  i  from  —oc  to  +00,  we  shall  deduce 
general  expressions  from  'p^  and  q^  arranged  according  to  the  power  of  the  eccen- 
tricity.    Since 

«  =  jf  -|-  e  sin  tt, 

•we  have  by  Lagrange's  theorem 


cos  V,  =  cos 


or 


g  —  esm'^fl ■'- ^ — etc.; 

''  ''       2\      eg  3!     fV 


'sm 


using  the  notation 


We  then  have 


C09  u^  —  2    -i  — -., 


n  +  l. 


n!  =  1.2.3 n=  r(H,  +  1). 

0!  =  1!  =  1. 


THE   ORIHT   OF    URANUS. 


16 


Siibstitutiiif,'  ill  tlie  gciicnil  term  of  the  above  scries  for  sin  g  its  value  in  imaginary 
cxpuuuutial  functions 

2  sin  flf  =  yiTl'  (o-»*^- »  —  cO^-\) 

we  find  by  the  binomial  theorem,  using  the  notation  of  combinations, 

«,_w(»  — 1) (»  — /,4-l)_         „! 

l.'i.JiVr.a'  "«!(»»  —  «)! 

I  n+l  il4 

'1+1  j 

Differentiating  n  —  \  times  with  respect  to  g,  and  putting  together  the  first  and 
last  terms,  the  one  after  the  first,  and  that  before  the  last,  and  so  on,  we  find 

_  2"+'  ^Tl'^^^'^^^'j}  =  (M  -(-  1  )»-i(„('.+i)(^'-r  ^  c-<"+"<' '  -0 
—  C(n  —  1)"-'  (c'— ""• -'  +  c-'"-')" '-"')  +  etc. 

n+l  1^1 

Substituting  for  the  exponentials  their  values  in  circular  functions,  and  dividing 
by  2"+'  we  have 


^"-'  sin"+V 
~  Og"-' 


-  L  {  (»  +  l;"-'  cos  («  +  l)g-  b{n  -  1)"-'  cos  (n  -  \)g 

*        I  n+l 

+  C  («  —  3)"-'  cos  («  —  3)  j7  —  etc.  I 

n+l  ) 

the  series  terminating  at  the  lust  positive  coeflficicnt  of  g.     Substituting  this  last 
value  in  the  general  term  of  the  series  which  gives  cos  «,  we  have 

cosM="F  ~^^^  |(«+l)-'cos(n+l)flr-r'(„_l)''-icos(«-l)^+ctc.   I 


Let  us  now  substitute  for  n  another  variable  /,  putting  in  the  first  term  of  the 
last  factor  t  =  n  -f  1,  in  the  second  /  =  ?i  —  1,  in  the  third  t  =  »i  —  3,  etc.  The 
limits  of  finite  integration  with  respect  to  t  will  then  be 

in  the  first       term,  -f-l  to  -\-<x., 

in  the  second  term,  — 1  to  -f-oc, 

in  the  third     term,  — 3  to  +oc, 

etc.  etc. 

But  all  the  coefficients  of  g  will  then  be  i,  and  the  formula  supposes  the  factor  of 
cos  ig  to  vanish  whenever  i  is  zero  or  negative;  whence,  those  elements  of  the 
finite  integral  in  which  i  is  negative  must  be  omitted,  and  all  the  terms  must  be 
taken  between  the  limits  +  1  and  +  oc.  Making  the  proposed  substitution  we 
have 


X8  THE  ORBIT  OF  URANUS. 

-,rt9 ,, V   J         ^ o<-i  * C'#>'+'_| : (7e'+* —  etc.  >  cos  lo 

''°'"-raci-l)!2'-''    -(l+l)!2'+»];/     +(i+3)!2'+»„*  j  "^ 

- 1.  (i-l) !  2^»  I         2^ i(t+l) <-.!.  +  2«i(t+l) (t-l-2) (i+3) *+«  i 

We  have,  therefore,  for  all  values  of  i  different  from  zero 

i'-«e'-»      jl       h        ^^       _J_^- —J^^' -ctclrU) 

'''=^-'=(i-l)!2--M         <+^"2^i^+lT"^*4^2*(i+l)(i+2)(i4-^  •;  '^ 

To  obtain  the  value  of  p„  we  remark  that  the  only  constant  term  in  cos  u  arises 
from  the  term  —  e  sin^  jr ;  its  value  is  therefore  —  i  e.  The  constant  term  in 
^  =  cos  u  —  e  is  therefore  |  e,  whence 

l)o  =  -3e.  (15) 

The  values  of  (ji  may  be  obtained  in  a  similar  way  by  developing  sin  «  by 
La  Grange's  theorem.  But  the  development  is  rather  more  complex,  and  it  is 
easier  to  derive  them  from  2>(.     Let  us  take  up  the  equations 

g  ^  cos  tt  —  e 

ri  =  sin  u 

u  —  e  sin  u  =  g 

Considering  m,  like  ^  and  >?,  as  a  function  of  the  independent  variables  e  and  «/,  we 
have  by  differentiation 

6u  _  d  (e  sin  w)  _  _ 

6e  Se         ~ 


du  _  d  (cy;)  _       sin  M 

'  de  ' 


6'e  1  —  e  cos  u 

1 


Comparing  («)  and  (I>) 


du  _  _______ 

dg       1  —  c  cos  u 

du       .      Su  d^du  6^ 

de  dg  dudg  dg 


dg  de 


(a) 


(i) 


Putting  in  this  equation  for  g  and  >?  thcii  developed  values  this  equation  becomes 

2  ipt  sin  ig  =2  -  \f^  -  sin  ig 
^  de 


which  gives  by  equating  the  coefficients  of  sin  ig 


(16) 


*T 


THE   OliniT   OF    URANUS 


n 


The  following  are  special  values  of  i\  and  g',,  developed  to  the  sixth  power  of 
the  eccentricities,  as  derived  from  the  preceding  formula; : 


2)o  =  —  3e 


11)2 


<y  —  -..V.  V,  e 


9216 


P^ 


•2" 
3 


.<^^  + 


IG' 


.J       45  5o7    J 

h''  ~128'^  +5120^ 


1,6, 

■^*  =  3^-ir/ 


Pi 


^^  =  m' 


125  ,_4;375  g 
27 


> 


Pt 
2^ 


16807 
46080' 

,       1 


8'  ^192         9216 


1 

^2' 
3 


1 


c»  + 


48 


,_  27    ,       243    , 
^8*-"      128''  +5120*' 


24-..'         1.-/ 


125 
^^"=384' 
27  . 


3125 


9216 


.  e" 


2" 


80 

16807 
'46080' 


(16)' 


Having  the  developed  ^  and  yi  in  terms  of  time,  let  us  resume  the  equation  (13). 
As  only  purely  linear  operations  arc  performed  on  Q  in  this  equation,  it  follows 
that  if  we  represent  its  several  parts  by  Q^,  (^.,  etc.,  and  by  (^p„  ^p^,  etc.,  the  values 
^p  obtained  by  putting  Q=z  Q^^  Q  =  Q^^  etc.,  we  shall  have 

^p  =  ^p,  +  (^p2  +  f  tC. 

We  have,  therefore,  only  to  find  the  separate  values  of  ro'5p  corresponding  to  the 
different  terms  of  Q,  and  to  take  their  sum.     Let  us  then  represent,  as  before,  by 


m 


any  one  term  of  R. 

3      April,  1873. 


"    h  cos  (iX  +  a  +/u'  +JL}) 


18 


THE   OUBIT   OF    URANUS. 


We  then  have,  considering  only  terms  of  the  first  order  with  respect  to  the  dis- 
tiirbing  forces, 

UtE  = sm  Jv, 


(17) 


fu,R  = 


m'ihv 


cos  N: 


•where  we  put  for  brevity. 


-n  ^  — 

cli  ,     a,         ,, 


1'=  -.;-,-,-— 


Let  us  represent  by  Q^  the  terms  in  Q  which  are  of  the  first  order  with  respect 
to  the  disturbing  forces,  so  that  we  have 

The  general  term  in  B,  will  tlien  give  rise  in  Q^  to  the  term 


m 


d 


a, 


cosN. 


a,      •     dx> 
In  the  case  of  the  action  of  an  outer  on  an  inner  planet  this  expression  becomes 


ai  V  '  d-of 


while  in  the  contrary  case  it  is 


—  ( 2tj7t  —  Ji  — *^  M  cos  N, 


both  derivatives  being  taken  with  respect  to  the  logarithm  of  the  mear  distance 
of  the  inner  planet. 

In  the  integration  it  will  be  more  convenient  to  substitute  for  ^'  and  X  the  mean 
longitudes  counted  from  the  perihelion  of  the  disturbed  planet.     If  we  put 


the  angle  N  will  become, 


Since  corresponding  to  each  set  of  values  of  i'  and  /  there  are  several  values  of/'  and 
y,  it  will  be  convenient  in  the  numerical  computation  to  combine  these  different 
terms  into  a  single  one,  because  after  forming  the  derivatives  of  R  there  is  no  need 
that  u,  <J  and  the  other  elements  should  appear  in  an  analytical  form.     If  we  put 


^'9 


THE  ORBIT  OP  URANUS. 


19 


Te  for  the  coefBcicnt  of  —  cos  N  in  the  preceding  general  term  of  Qa,  this  term  will 

become 

g,  =      —  h  cos  [/g>'  +  {i!  +  i  +i)  u]  cos  [tT  +  ig\ 
a, 

_  —  k  sin  [iV  +  (i'  +  i  +j)  w]  sin  [t7'  +  ig-\ 
«i 
If  we  put 

h  =  2  i  cos  [  j'J  +  (»•'  +  i  +y)  o], 
k.=Xlc sin  [j'o'  +  (*'  +i+j) u], 

the  sign  2  being  extended  so  as  to  include  all  values  of  J  and  f  which  correspond 
to  tlie  given  values  of  i  and  i',  we  shall  have  for  the  general  terms  of  Qa 

^  I  h  cos  (*Y  +  ii/)  +  /.-.  sin  (lY  +  if/)  I , 

or,  when  we  represent  the  angle  i'l'  -\-  i<j  by  JV, 

(2„  =  ^''|/.-„c.siVi  +  A-;sinJ\^,|. 

This  we  are  to  combine  with  the  values  of  !^  and  yi 

y;  =h^qj  sin  jg, 

in  the  general  integral  formula  (13).  If  we  substitute  them  in  this  formula,  and 
represent  by  ft  the  coefficient  of  t  in  the  value  of  N  we  shall  have  to  integrate 
differentials  of  the  form 


sm 
cos 


(iVi±*i/) 


in  which  the  coefficient  of  the  time  t  in  the  angle  is  /t  +  in.     Let  us  represent  by 
V,  the  integrating  factor 

[i-\-m' 
The  formula  (13)  will  become  by  these  substitutions,  which,  though  a  little  comr 
plex,  offer  no  difficulty, 

,         1      m'arr'   +? 

^<'  =  16a.(i+,«-f|>-^^'^X 

I  i'+j-v+,\  1/h.cos  [N,-\-{i+j)g-\+k,sin  [Ar,+(i+y)^]J 
+ } r+t-r-j \  \ A-oCos  [A\+{i-j)g]-\-k, si"  [iV.+('-i)i/] \ 
+ 1  .'+,->'_, \  \ h cos [iV,-(i-y)«/]+A-.sin [iV;_(/-j)»/] \ 
-\-\v.,—v_j\  lAvcos[iVi-(i+y);7]+A-.sin[Ar,_(t+y)i/]| 

The  sign  2  of  finite  integration  here  includes  the  separate  combination  of  every 
value  of  i  with  every  value  of  j,  except  those  combinations  which  make  the 


♦  Tlio  imlloes  ?  and  j,  in  tlipsn  eqnatidns,  arc  not  to  be  ponfoundcd  with  the  coefficients  of  x  and  u 
In  the  general  terms  of  R  and  <^.     We  need  not  use  the  latter  at  present. 


ta!^ 


i 


20 


THE   ORBIT   OP   URANUS. 


coefficient  of  tfie  time  under  the  sign  mn  or  cos  vanish,  and  so  render  the  corre- 
sponding value  of  V  infinite.     These  cases  have  to  be  treated  separately. 

To  find,  from  the  expression,  the  coefficient  of  the  sine  or  cosine  of  cosine  Ni-\-ug 
in  r,^Sp,  we  put,  in  the  four  lines  of  this  equation,  e  :  follows: 

In  the  first,  i-\-j=u.'.j=      u  —  i; 

"       second,      i — J=:u.'.j=       i — u; 
"        third,    — i-\-j=u.\j=       u-\-i; 
"       fourth,  —  i  —j  =  u  .•.  j=  —  u  —  i. 
In  the  above  expressions  i  and  j  being  independent,  and  including  all  values 
from  —  oc  to  -foe,  »■  and  u  will  also  be  independent,  and  include  the  same  range 
of  values.     Substituting  for  J  its  value  in  it  the  coefficient  of 


VI  a  r 


becomes 


16a,(l-f-m) 


[k,  cos  ( A^i  +  itg)  +  k,  sin  {N,  +  ug)} 

+ i>i  ?-(«+()  (''-(  —  »'(«+.■))• 

Since  q^  =  —  q^j  this  expression  reduces  immediately  to 

"  \  +  J'. '?.■+»)(»'("+.)— »'-i    )' 

or,  substituting  t  —  u  for  i  in  the  second  line 

22  (j',?(,-„)  +  i',(-.)?()  (''( —  »'«-()• 
Hence,  writing  N  instead  of  Ni, 

•      8ai(14-w») 

This  expression  fails  for  the  particular  case  iV=  vg,  where  the  value  of  j'_„  will 
be  infinite.     If  we  take  each  term  of  Q  of  the  ibrm 


m 


(Ai">  COS  vg  +  A;*"'  sin  vg). 


and  substitute  in  the  general  expression  (13)  it  will  be  found  that  the  terms  in  r^-fip 
which  have  the  infinite  values  of  v  as  a  factor  are  to  be  omitted,  and  replaced  by 


^       *a,(l  +  wi)" 


(20) 


The  two  parts  of  i-j'Sp  thus  found  include  all  the  terms  of  the  first  order  with 
respect  to  the  disturbing  forces.  But  when  terms  of  the  second  order  are  taken 
into  account,  we  shall  find  terms  in  Q  proceeding  from  secular  variation  in  wliich 
the  time  appears  as  a  factor,  outside  the  signs  sin  and  cos.  Let  us  represent  such 
of  these  terms  as  depend  on  any  angle  N  by 

Q  =  V}^±  (k,  cos  N-\-k,  sin  N) 


* 


THE   ORBIT  OF  URANUS. 


31 


; 


and  use  the  symbol  j'j,  as  before,  to  represent  the  ratio  of  the  mean  motion  of  the 
phinet  to  the  coefficient  of  t  in  the  angle  ^-|-  i<j,  so  that  if  fi  represents  the 
coefficients  of  t  in  N  we  have 


n 

n 


Vi  = 


^'+  in       fi' 


n 


1 


2'i',<7.  -i 


we  find  the  expression 
G Qdt=^  ]  "''^^'-  \  (r,V.-„  -  vAnt) cos  {N+  i,j)  +  (vi,  h -  V-Ant)  cos  (iV  -  tg) 
(r.V.-.  +  vAnO  sin  (.V+  /r/)  +  (vi,  t.  +  r_(7.-,.«0  si"  C^"-  «i/)  J 

r,?gf?<  =  ]  '"-*  I  (r//.-.  -  vjhnt)  sin  (iV+yy)  -  (vljh  - 1^-/.-.«0  sin  (iVT-jV,) 

-  ('■;-/'•.  +  rjhnf)  cos  (iV+yy)  +  (vi/-,  +  v.^wO  cos  {N-jy)  J  . 
If  we  now  put  for  brevity 

)•("/.•„  —  !■(/.•.«<  =  s„ 
the  general  value  of  r^^l^p  becomes 

1      m'arT^ 

^^'-VG  «,(!  +  "') 

(<•    -  c,  )  cos  (.V+  {i  +/)  <7)  +  («<   -  «;  ) sin(xV+  (t  +y)  </)  " 
+  ('■<  -  C-.)  cos  (.V+(t-i )(/)  +  («_,-«,  )sin(.V+(;-y)gr) 
+  (c,.  —  c_,)  cos  (.V—  (i  — y)  fj)  +  (Af_,  —  s^  ) sin (.V—  (i—j)  (j) 
,  +(c_,-c_,)cos(.V-(i+y)i/)  +  (s_,-«_,)sin(i\r_(t+y)j/) 

If,  as  before,  we  transform  this  expression  by  putting 

in  the  first  line  j  =^u  —  »  ; 
in  the  second  "  y  =  t  —  u; 
in  the  third  "  j—i-\-  u ; 
in  the  fourth     "      j  —  l—u- 

the  value  of  r{'8^>  reduces  to 

1       «»'« 

is  a,(l+J/0  ^ 
V  2   f  i'.?(u-o  (C..-0-  c.  )  cos  (iV+  «^)  -!  i),7„_„  («<-  «,.-„)  sin  (iV+  w^/)  ) 
"'"'  li'i'Z(a4.)(c,„,. )-"-<) cos (.V+«j/)  -ri'*,.H)(-'~V+o)s»n(^V+«i/)  i 

or,  putting  t  —  u  for  i  in  the  last  line, 
1     m'arT^ 

^  2„^,  1  i'/Au-o  -i'(,.-,,7, !  S  (c>„-.)-  '■.)  cos  ( A^+  vy)  +  («.  -  «„_.)  sin  (A^+  wjr)  ( ; 


HI 


2'i 


THE   ORUIT  OP   URAXUS. 


to  which  expression  is  to  be  added,  in  lieu  of  tlie  terms  which  will  have  iufinite 
values  of  p  as  a  factor. 

1  m'ar-hi-e 


^aJi^^^^-^-^^^^^n 


(22) 


for'  "o^  ^'''  ^""^  *^'*'  ^'''''''''  °^  «.  "' ''°'  "^  *""*  a]  "'  ''"  "^  ^"  *''^  expression 
The  formulae  19,  20,  21,  and  22  give  the  complete  expressions  for  the  perturba- 
tions of  the  logarithm  of  radius  vector  by  successively  substituting  in  it  all  tho 
terms  of  Q. 

Perturbations  of  Longilade, 
We  now  pass  to  the  perturbations  of  longitude.  In  the  Mecanique  Celeste 
(Premiere  Partie,  Liv.  ii.  Chap,  vi.),  Laplace  gives  an  equation  (Y)  by  which  the 
perturbations  of  longitude,  which  are  of  the  first  order,  may  be  derived  from  those 
of  the  radius  vector  without  the  formation  of  any  other  derivatives  of  i?  than  those 
which  enter  into  Q.  But  tlie  formula  does  not  seem  easily  adapted  to  the  case  in 
which  the  perturbations  of  the  second  order  are  taken  into  account,  we  shall 
therefore  derive  all  the  perturbations  of  longitude  from  the  second  of  equations  (1) 
liy  integration  this  equation  gives 


do      ft  {  rSE  ,  ) 

dt=r"-\J6y<^(+(^\ 


IS 


0  being  the  arbitrary  constant  of  the  integral,  llepresenting,  as  before,  by  sub- 
scnj.t  zeros  tlie  values  of  the  co-ordinates  corresponding  to  the  ellipse  to  which  the 
orbit  IS  supposed  to  reduce  itself  when  the  disturbing  forces  vanish,  we  have 

fA'o n"-n  cos  i//       fiO 

because  the  constant  to  which  the  integral  must  reduce  itself  in  the  elliptic  motion 
a^>i  cos  T^       o  w      i- 

-  ~-      Subtracting   the   last   equation  from  the  preceding,  and  putting 

V  —  I'o  =  Sv,  we  find 

dhv  _  ,1   rdRj       /I        1  V 

1 

Developing  -^.^  to  terms  of  the  second  order  with  respect  to  the  disturbing  force 

-^  =  ,\(l-2<V  +  2V-etc.), 
which,  being  substituted  in  the  last  equation  by  putting 


/'  = 


gives 


.d^v 


r,'        = 


1+w' 


-SR 


' an  ra/i 

which  is  r.,;   ous  to  quantities  of  the  second  order. 


(23) 


J 


TUE    ORBIT   OF   UUANUS. 


33 


1 


The  most  convenient  mode  of  making  the  numerical  computation  of  the  second 
order  terms  by  means  of  this  equation  will  depend  upon  circumstances.  If  the 
perturbations  of  longitude  and  radius  vector  of  both  planets  are  already  known  with 
a  sufficient  degree  of  approximation  for  the  computation  of  formula  (11),  it  will  be 
more  convenient  to  form  at  once  the  complete  values  of  all  the  quantities  which 
enter  into  the  equations  (12),  (13),  (19)  to  22),  and  (23),  so  that  no  steps  of  the 
process  shall  have  to  be  repeated.  If  such  perturbations  are  not  known,  they 
must  first  be  computed,  and  it  will  then  be  necessary  to  begin  with  the  perturba- 
tions of  the  first  order,  and  afterward  add  those  of  the  second.  There  is,  how- 
ever, one  class  of  terms  of  the  second  order  which  it  will  be  most  convenient  to 
take  account  of  from  the  beginning,  namely,  those  arising  from  the  constant  term 
in  fip  and  Sp.  This  is  eff'ccted  by  correcting  the  mean  distances  for  an  approximate 
value  of  these  constants  at  the  beginning  of  the  computation,  and  then  proceed- 
ing in  the  usual  way.  This  is  in  fact  what  we  have  supposed  to  be  done  in  the 
preceding  investigation.  The  values  of  6v,  hV,  hp,  hf>'  in  formula  (11)  will  then 
contain  only  periodic  terms. 

In  computing  the  terms  of  the  first  order  we  determine  the  value  of  5p  from  the 
equations  (19)  and  (20),  using  the  value  of  ^o  in  il»).  '-I'l^en  those  of  tv  are 
obtained  by  integrating  the  equation 


lit 


=  l  +  m«^   ^v'^'-2"^°«^r.'- 


(24) 


Having  found  tlie  values  of  hv  and  Sp  for  both  planets,  they  are  to  be  substituted 

in  (11),  to  obtain  5^,  5-^^and  h  ^!^.      But,  rigorously,  Sv  and  ^v'  arc  not  the 

6v  op 

same  with  So  and  M,  owing  to  the  movement  of  the  orbits  of  the  planets,  and  the 

corrections  for  Sy  are  also  to  be  added.     Considering,  for  the  present,  only  the 

perturbations  of  the  second  order,  which  depend  on  Iv,  ^v',  Sp,  and  5p',  we  may 

use  the  following  equation  for  hE,  and  similar  ones  for  its  derivatives: 


s«  ='"*»+ '5  w+ 


vR 


m 


^p+'^-^-.^p', 


6\'  ^"'^  <?p    '    '    dp' 
Having  thus  found  ?fR,  and  hence  D'tf'B  by  differentiation,  and  then  S 

6R       l<r(;-„V)    ,    I'V 


the  quantity 


6R 

'6p 


(2;3) 

,  wo  form 


'0 


(2G) 


/^   dii       1  '^"('•o"'V") 
D',>RcU  +  8   ^^-^^      </<^' 

which  is  the  difference  between  the  value  of  <?«  i»  (1*^)  and  that  of  Q  in  (12). 
The  terms  in  ^p  arising  from  bQ  are  then  to  be  computed  by  tlie  formulic  (19), 
(20),  (21),  and  (22),  when  we  shall  have  ^p  accurate  to  quantities  of  the  second 
order.  Let  us  represent  these  additional  terms  by  d^p.  Subtracting  (24)  multi- 
plied by  r,*  from  (23),  recollecting  that  the  ^p  which  appears  in  the  second  term 
of  the  "former  is  really  hp—hy  we  find,  neglecting  quantities  of  the  third  order, 


idb 
dt 


'«  =  «n»  |/S  •^^  dt  -  2^p/^^^»  dt  I  -  2«  cos  ,j.  (,^»p  -  5p«) 


24 


THE   ORDIT   OP   URANUS. 


from  whicli  the  terms  of  5i'  of  the  second  order  arc  obtained  by  multiplying  by 


ri~'^  and  integrating. 


Motion  of  the  Orbital  Planca. 

The  general  theory  of  the  motion  of  the  planes  of  reference,  especially  of  the 
motion  of  the  instantaneous  orbit,  has  been  so  often  treated  that  I  can  scarcely 
hope  to  add  anything  essentially  new  to  it.  1  shall,  however,  endeavor  to  pre- 
sent the  differential  equations  of  tlie  motion  in  a  simple  and  general  form,  and 
one  in  which  the  geometrical  conceptions  of  the  problem  shall  be  made  as  clear 
as  possible. 

The  orbital  plane  of  each  planet  being  at  each  moment  osculatory  to  that  part 
of  the  orbit  which  the  planet  is  actually  describing,  its  only  motion  is  one  of  rota- 
tion around  the  radius  vector  of  the  planet  as  an  instantaneous  axis.  This  rota- 
tion may  be  resolved  into  two  others  around  any  pair  of  rectangular  axes  fixed  in 
the  moving  plane.  But  the  rotation  produced  by  any  one  planet  is  most  sim])ly 
expressed  when  inferred  to  axes,  one  of  which  coincides  with  the  common  node  of 
the  two  orbit"*.  The  rotation  produced  by  each  separate  planet  must,  therefore,  be 
first  referred  to  its  node  on  the  moving  orbit,  and  then  the  combined  rotations 
must  be  resolved  into  two  aroinid  axes  assumed  at  pleasure.  To  effect  this,  let  us 
suppose  positive  rotation  around  an  axis  to  be  such  that  an  observer  looking  from 
the  origin  along  the  positive  direction  of  the  axis  sees  the  right  hand  side  of  the 
plane  move  downwards,  and  the  left  hand  side  upwards.  Let  us  also  denote  the 
first  axis  in  the  order  of  longitude  the  principal  axis,  or  that  of  A'  and  that  90° 
farther  advanced  the  secondary  axis,  or  that  of  Y.     Let  us  now  put 

ilq,  the  instantaneous  rotation  around  the  axis  of  X; 

dj),  the  instantaneous  rotation  around  the  axis  of  Y.  Let  us  also  put,  relatively 
to  any  disturbing  planet, 

d/;,  the  instantaneous  rotation  around  the  ascending  node  of  the  disturbing  planet 
on  the  orbit  of  the  disturbed  one. 

die,  that  around  the  corresponding  secondary  axis. 

Then,  from  the  known  equations  for  the  perturbations  of  the  inclination  and 
node  of  an  orbit,  we  find,  that,  if  any  term  of  the  perturbative  function  be  repre- 
sented, as  before,  by 

"''*  cos  (i'A'-l-a-f/u'+y«), 


the  differential  rotations  y;  and  Jc  will  be  given  by  the  equations 
dy;  m'h 


dt 

die 
dt 


an 
cos 


T  {  (*'  +J)  coty  -j-  (t"+/)  cosec  y  Um  iV 


711  an 


cVt 


cos  N. 


<f,  cos  il/dy 

As  E  is  actually  developed,  the  mutual  inclination  y  does  not  explicitly  appear, 
but  is  replaced  by 

a  =^  sin  1  V. 


*»*  » 


;.  -tuv  .  -?"  ■'aij^ff*  *-aK 


THE    ORBIT   OF   URANUS. 


90 


Milking  this  substitution,  and  putting  iilso 

»  +  *+/+y=  — '• 

these  equations  become 

at      «,  COS  4'  cos  ^y  {  2a  '        '  ) 

lU'  m'lm  COS  \y  Ch        ,» 

,    =—     ,   "        ,  '   „    cosiV. 
ut  2a,  (08  -^     do 

To  pass  to  the  general  rotations  </j>  and  dq,  lot  us  represent  by  0„  0,j,  etc,  the  lon- 
gitudes of  the  ascending  nodes  of  the  several  orbits  of  tlic  disturbing  planets  on 
that  of  the  disturbed  planet.     We  shall  then  have 


dq_^ 
dl 


dr, 
dt 


ScosO,  "'"  —  2  sinO, 


dl 


dp       —         „  dh\   ,  „    .     ^  dr, 

,,^=scoso<^;+vsmo,^y;. 


(28) 


These  equations  completely  define  tlic  instantaneous  motion  of  the  orbital  plane. 
They  cannot,  however,  be  rigorously  integrated  in  their  present  form  because  p 
and  q  as  integrals  have  no  completely  defined  signification.  To  do  this  it  is  neces- 
sary to  express  the  difttrential  rotations  dp,  dq,  etc.,  in  terms  of  the  differentials 
of  any  elements  wo  may  select  to  define  tlic  position  of  tbe  orbital  plane,  and  then 
to  integrate  the  equations  thus  formed.  But,  for  the  purpose  of  constructing  tables 
of  the  planets  we  may  consider  ^),  q,  etc.,  to  represent  small  rotations  of  the  planes 
of  which  the  powers  and  products  may  be  neglected,  and  the  integration  is  then 
quite  simple. 

Peiiitrhadons  of  the  second  order  depending  on  the  motion  of  the  orhital  planes. 

R  being  a  function  of  the  five  quantities  of  r,  r\  v,  v',  and  y,  the  motion  of  tlie 
orbital  planes  introduces  tcrnis  of  tiic  second  order  by  clianging  the  values  of  v,  v', 
and  y.  These  terms  we  liavc  hitherto  neglected.  To  investigate  them  let  us  refer 
the  rotations  of  both  planes  as  given  by  (28)  to  the  node  of  the  disturbing  on  the 
disturbed  planet  as  the  principal  axis.  If  wo  represent  by  dy;,dK;d/;\  and  (//.;'  the 
rotations  corresponding  to  this  axis,  and  designate  by  the  subscript  1,  the  quantities 
which  refer  to  the  disturbing  planet  whose  action  we  are  considering,  and  by  2,  3, 
etc.,  the  other  planets,  the  equations  (28)  will  be  replaced  by  these 


S' ---(«.-".)  *■• 


•».+v, !„(„,_„,)*., 


the  summation  commencing  with  /  =:  2. 

By  formuhc  of  tbe  same  kind  wo  arc  to  find  the  difl"erential  rotations  dy;'  and 
dk'  of  the  orbit  of  the  disturbing  planet,  produced  by  the  action  of  all  the  planets. 


4      April,  1873. 


Iw  ^ 


96 


THE   ORBIT   OF   URANUS, 


These  rotations  will  be  nroimd  tlio  snmo  principal  axis  with  the  rotations  dn  and 
*//.-,  but  around  a  secondary  axis  in  the  piano  of  the  disturbing  orbit,  and  therefore 
making  an  angle  y  with  the  secondary  axis  of  the  disturbed  orbit.  A  geometrical 
construction  will  now  sliow  (p.ite  simply  that  the  infinitesimal  rotations  hn,  hk,  bn\ 
and  hic'  will  produce  the  following  changes  in  v,  v',  and  y. 

^v  =  cot  ybk  —  cosec  yhK 

hV  =  cosec  yU-  —  cot  yhk '  (Op) 

hy  =  hri —  A»7 

If  wo  substitute  these  values  in  the  general  fornndic  (11)  the  terms  of  the  second 
order  added  to  hR  will  be 

(6R     .         dR 


SD      /o'^      .  SR  \ 

\6v  ^°^  ^  +  <5v'  ^°^^^  ^'r^ 

-(^^cosccy+   ,^cotypk^ 


(30) 


^k 


The  first  two  terms  of  this  expression  may  be  put  into  the  form 

{i(2f  +  ^:;)(— y+eoty)-j('^f__;^;?_)(._^_,,,^)| 

-  {  K  7!  +   t^)  (^"^^'^  y  +  cot  y)  +  I  (  f^    -  I'JL)  (eosec  y  -  cot  y)  I  hk. 
But, 

cosec  y  +  cot  y  =  cot  i  y  =  ^^^  ^  y, 

a 

cosec y  —  cot y  =  tan  ^y=       " 

cos  i  y 


and  in  the  general  term  of  R,  by  (7) 

rR m'h 

Ox  ~        a, 


0'+y)sin-Ar 


Making  these  substitutions,  and  putting,  as  before, 

the  above  value  of  SR  reduces  to 
m'h 


,    w/cosjy  clh    .  ,       . 


(31) 


,WJ 


THE   OllUIT   OF    URANUS. 


9t 


1! 


Tlio  currcsnoiiiliiig  tLriim  of  h         and  h  ',    <  uiul  may  ho  ulitaiiicd  in  tliv  satiie  way 

liy  Miibntituting  ^,     uud  '[      tur  A'  iu  (:3())  uud  coutiuuing  the  correspuuding  sub- 

stitutions  of  the  general  terms  of  tlie  derivatives  of  It  as  given  on  page  9. 

Tlic  equation  (31),  besides  being  of  tlie  second  order  witli  respect  to  the  disturb- 
ing forces,  is  also  of  tlie  second  order  with  respect  to  tlie  mutual  inclinations.  For 
Ms  hkt  hr„  and  hy;  arc  of  the  first  order  with  respect  to  both  quantities,  and,  when- 
ever (  is  not  zero,  h  is  a  quantity  of  the  second  order,  containing  a^  as  a  factor.  It 
is,  therefore,  only  in  exceptional  cases  that  the  terms  of  th(!  second  order  depend- 
ing on  the  motion  of  the  orbital  planes  can  become  sensible. 

JiediwIuM  of  (lie  loiigUiiJe  in  the  or/til  to  lo  tjitude  on  (he  ecl!j)lic. 

The  integration  of  (23)  gives  a  value  of  ftv,  which,  added  to  the  longitude  iu 
orbit  corresponding  to  tlie  pure  elliptic  motion  gives  the  longitude  in  the  disturbed 
orbit,  counted  from  a  fixed  point  in  tiie  moving  plane  of  that  orbit.  The  position 
of  this  fixed  point  is  comi)letely  determined  by  the  condition  that  the  instanta- 
neous rotation  of  the  plane  in  question  around  tiie  axis  perpendicular  to  itself  is 
always  zero,  so  that  the  motion  of-the  point  of  reference  is  always  perpendicular  to 
the  direction  of  the  plane.  IJut,  altliougli  tliis  instantaneous  rotation  is  zero,  the 
integrated  rotation  is  not  rigorously  zero  when  W(.>  consider  tin;  terms  of  the  second 
order.  It  follows  that  the  value  of  r,  the  longituoe  in  orbit,  and  the  position  of 
the  piano  of  the  orbit  do  not  rigorously  determine  the  position  of  the  planet:  we 
must  also  know  how  the  fixed  point  of  reference  has  changed  its  position  iu  con- 
fiequence  of  the  motions  which  the  plane  has  undergone.  Let  us  consider  the 
relative  positions  of  this  plane  at  two  epochs.  If  the  fixed  point  were  equally 
distant  from  the  common  node  of  the  two  planes,  the  integrated  rotation  of  the 
plane  around  its  own  axis  would  be  zero.  But,  these  distances  not  being  equal, 
tiieir  difference  is  a  correction  to  be  ajiplied  to  the  longitude  of  the  planet  in  its 
orbit.  Suppose,  now,  that  at  the  end  of  any  time  the  inclination  of  the  actual 
orbit  to  tlie  primitive  orbit  is  <^,  and  the  distance  of  its  ascending  node  from  the 
present  position  of  the  moving  axis  of  a;  is  0.  A  rotaticm  around  tiu'  line  of  nodes 
will  not  change  the  quantity  sought.  But,  if  we  represent  the  infinitesimal  rota- 
tion around  an  axis  perpendicular  to  it  by  dr  we  shall  have 

cos  0  dp  —  sin  0  dif  =;  di\ 

dq  and  dk  being  the  instantaniMins  rotations  around  th(>  resp(>ctive  axes  of  a*  and  y. 
By  this  rotation  it  is  easy  to  see  that  the  relative  distance  of  any  two  fixed  points, 
one  on  each  plane,  from  the  node,  will  be  altered  by  the  quantity, 

dr  (cosec  ^  —  cot  ^)  =  dr  tan  \  <^, 

the  relative  longitude  of  the  fixed  jioint  on  the  moving  plane  being  increased  by 
this  amount.     Tiio  correction  to  the  longitude  in  orbit  from  tliis  cause  is,  therefore, 

(//  —  dr  tan  J  </)  =  tan  ^  ^  (cos  6  dp  —  sin  0  dq). 


' 


28 


THE   OIIHIT   OF   U  11  AN  US. 


Counting  the  integrated  values  of  p  nnd  q  \\\  a.  tlirection  perpendicular  to  the 

moving  plane  wo  iiiivu 

„       taiii) 
sin  0  =       ' 
tun  <p 

cos  0  =        ' 
tun  <p 

which,  heing  substituted  in  the  expression  for  dl,  gives 


(11: 


COS(^ 


(tiui  qdp  —  tun  j>f/(/). 


(32) 


1  -f-  cos  tp 
The  approximate  value  of  the  integrated  correction  is  therefore 

5^  =2 /('/'(/'— My)- 

For  every  pair  of  periodic  terms  in  jt  and  </,  such  as 

q  =  8  sin  fit,  p=.8  cos  fit, 

U  will  contain  the  secular  term  —  J  /*'  fit,  which  will  be  confounded  with  tlie  mean 
motion,  and,  if  it  were  not  so  confounded,  would  in  few  or  nono  of  tlie  larger 
planets  amount  to  u  second  in  a  thousand  years.    If  the  secular  terms  in  p  and  q  be 

q  =z  at  ;  2)  =^  n't 

f.l  will  vanish.  We  hence  conclude  that  these  terms  arc  entirely  unimportant  in 
the  present  state  of  astronomy,  and  that,  if  we  consider  the  positions  of  the  plane 
of  the  orbit  at  two  epochs,  we  may  consider  the  points  of  departure  in  them  to  be 
equally  distant  from  their  common  node. 

We  have  therefore  only  to  consider  the  motion  of  the  inclination  and  node  duo 
to  the  change  of  the  position  of  the  orbit  and  of  the  ecliptic.     If  we  put 

^,  the  inclination  of  the  orbit  of  the  planet  to  the  cclipt'", 

0,  the  longitude  of  its  node  counted  on  the  ecliptic, 

T,  the  longitude  of  the  same  node  counted  from  tlie  same  -  '  -'niut  in  the 
moving  plane  of  the  orbit  from  which  v  is  counted. 

Then,  the  longitude  of  the  planet  on  the  ecliptic,  or  L,  will  be  given  by  the 
equation 

tan  (  L  —  0)  =  cos  ^  tan  {v  —  t), 

or,  when  developed  in  powers  of  <^, 

L  =  v-\-e  —  r  +  D,  (33) 

where  D  is  the  reduction  to  the  ecliptic,  the  value  of  which  is 

D  =  —  tan^  J  4)  sin  2  {v  —  t)  +  J  tan'  |  <^  sin  4  («  —  t)  —  etc. 

Let  us  refer  the  instantaneous  rotations  of  the  orbit  and  of  the  ecliptic  to  the 
fixed  points  of  reference  in  the  two  planer ;  q  being  the  rotation  around  an  axis 
passing  through  the  sun  and  the  fixed  point,  and  p  that  around  an  axis  in  90° 
greater  longitude,  and  the  accented  quantities  referring  to  the  ecliptic.  We  then 
have 


t;- 


V^J 


m^waamiieitmm 


' 


» 


Til  K   Oil  HIT   OF   URANUS. 


99 


'  —  I* 


'/o 


ill 


=  co8T  .  -f-sinr 


■cosO 


lit 
d<f 

dt  ' 


■  sin  0 


dp 

dt 

dp' 

dt 


dL 
dt 


dt  =  '•°'"'*''''  (-'■"  ^  'dt  +  '°'^  '?<' )  ^®*^ 

+  cot     <^(     sinO'^^'f-cosO'J;) 
dr         ^       I      •      ih    ,  ''/'  \ 

4-cosec<Ji(      sinO    .'  — cosO   '    ) 

If  wc  dift'ercntiiite  (;};3)  ami  substitute  these  values  of       and      ,  we  shall  have 

dt  dt 

dv     ,    dD       ^       .      /  dp  .        dn    ,  „  <///         .    «  </'/ \         /.i.-\ 

dt  +  ^^/^- tun  i.^  (cost  J^  -«mr-^^;  +cos0  j^  -^mO--^;  )        (J-O 

If  wc  consider  only  quantities  of  the  first  order  with  respect  to  the  disturbing 
forces,  wc  may,  in  integrating,  suppose  t  and  0  equal  and  constant,  and  (p  constiuit. 
The  integral  will  tlieu  be 

A  =  ('  +  i>  +  tan  i  <^  S  cos  0  {hh  +  W)  —  sin  0  ((V  +  'V)  \  (3fi) 

In  the  case  of  Uranus,  tan  <^  is  so  small  that  this  equation  will  be  sufficient  for 
a  long  tmie  before  and  after  our  epoch. 

In  the  application  of  the  method  to  other  planets  the  mode  of  operation  must 
depend  on  the  circumstances  of  each  particular  case.  The  differential  equations 
(34)  between  0,  t,  and  <^  are  rigorous,  and  their  integrals  may  be  approximated  to 
in  various  ways,  out  of  which  that  best  applicable  to  the  particular  case  must  be 
selected. 

Expressions  for  the  latitude. 

If  the  position  of  the  orbital  plane  and  of  the  ecliptic  were  each  determined  by 
tlie  i)receding  formuUr,  there  would  be  no  perturbations  of  the  latitude,  the  lati- 
tude itself  being  given  rigorously  by  the  equation 

sin  [i  =  sin  ^  sin  («  —  t). 

=  sin  (J)  cos  T  sin  v  —  sin  ^  sin  t  cos  v. 

But  the  instantaneous  values  of  ^  and  t,  or  of  sin  ^  cos  t  and  sin  <^  sin  t,  are 
troublesome  to  tabulate ;  it  will  therefore,  in  practice,  be  found  mor"  convenient  to 
use  only  their  mean  values,  and  to  consider  their  changes  from  this  mean  as  per- 
turbations of  the  latitude.  Representing  by  the  sign  h  the  deviations  from  tiie 
mean  values,  which  are  of  course  arbitrary,  we  have 

cos  /35/i  =  cos  <^  sin  {v  —  t)  5(^  —  sin  <^  cos  (v — t)  hr. 

Let  us  substitute  for  S<^  and  ^t  their  values  given  by  the  integration  of  (34)  to 


lJ 


80 


THE   ORBIT   OF   URAXUS. 


quantities  of  the  first  order,  in  which  case  0  and  t  may  be  assumed  equal.  Tliese 
vahies  are 

h<p  =  sin  T?p-\-  cos  r  ^q 
blii  (pbr  =^  cos  <^  (cos  T  <^p  —  sin  t  ^q) 

the  terms  dependent  o;i  ^j>'  and  f.q'  being  omitted  because,  I'cing  purely  secular, 
they  may  be  included  in  the  mean  values  of  <^  and  t.  Substituting  in  tlie  expres- 
sion for  S^ 

cos  ;3<V3  =  cos  ^  I  sin  vlq  —  cos  vlp\.  (37) 

In  the  case  of  all  the  larger  planets  both  cos  (i  and  cos  ^  may  here  be  put  equal 
to  unity,  when  the  expression  for  h^i  will  become 

hli  =  sin  V  hi  —  cos  v  }p.  (38) 

To  develop  this  expression  in  purely  periodic  terms  we  must  substiiute  for  v  its 
value  in  terms  of  the  mean  longitude  or  mean  anomaly,  namely, 

V  =  Z  +  2e  sin  jr  -|-  ,  e*  sin  Ig  -j-  etc. ; 

suppose  the  terms  of  ;^j)  and  Iq  depending  on  any  argument,  N  to  be 

^^  =  —  a^siniV— «,.cos  JV  /jjqn 

hq^=      a',  sin  iV-|- «',.cos  .V 

and  put  71  for  the  longitude  of  the  perihelion,  so  tliat 

then,  to  terms  of  the  first  order  witli  respect  to  the  eccentricities,  we  have 
5;i  =  —  t;  (ff,  cos  7t  +  «',  sin  7t)  sin  iV— e  (a,  cos  71  +  «'„  sin  7i)  cos  iV 

+  \  S((i,  +  «'e)cos7t  +  K  — «e)sii»7i|  sin(.V+  O) 
+  \  \  ("c-  —  O  i-os n  +  («'o  +  <'J  sin  n \  cos  {N-\-  y) 
_|_  1  J(a, —  «',)  cos 7t +(«',  +  ",  1  sin  7t I  sin  (.V—  <j) 
-i-i  S(«„  +  a',)cos7r-i-(«'e  — 't.)sin7iicos(.V—  g) 
+  h'']  («.  +  «'c)  f OS 71  +  («',  —  «,.)  sin 71  \  sin"  ( .V-f-  2r/) 
+  ^e)  (a,  —  a',)  coSTt  4-  («'-■  +  ".I  siuTil  cos(.V+  'ig) 
+  ^  e  j  (a,  —  a',)  cos  7t  +  («'«  +  «,.)  sin  7r  |  sin  (iV  —  2.7) 
+  i  e|  (rt„  +  O  cos 71  +  K  -  fl.)  sin  7t|  cos  (iV— 2*7) 

The  point  of  the  orbit  from  which  n  and  v  are  counted  is  entirely  arbitrary, 
and,  in  considering  the  action  of  but  a  single  planet,  it  will  he  most  convenient  to 
count  them  from  the  common  node,  in  wliich  case  n  must  be  replaced  by  o,  and 
fp  and  <^q  by  ^,k  and  fty;.  Thus,  'educing  the  perturbations  of  the  latitude  imme- 
diately from  the  formulje  (27),  we  shall  have 

6/i  =  sin  v  5)7  —  cos  V  (^A. 


(40) 


I 


THE   ORBIT   OF   URANUS. 


81 


CIIAPTEIl    II. 

APPLICATION  OF  THE  PRECKDING  METHOD  TO  THE  COMPUTATION  OF  THE 
PERTURBATIONS  OF  URANUS  BY  SA.TURN. 

Data  of  Conqnitaiion. 

The  elements  of  Uranus,  adopted  in  this  computation,  were  deduced  from  the 
comparison  of  nine  normal  heliocentric  longitudes  at  intervals  of  3(597  days  extend- 
ing from  1781,  December  26,  to  1862,  December  18,  with  corresponding  provisional 
places  derived  from  the  elements  given  in  the  "  luvcsitigation  of  the  Orbit  of  Nep- 
tune," with  perturbations  produced  by  Jupiter,  Saturn,  and  Neptune.  As  the 
perturbations  are  to  be  entirely  re-computed,  and  the  elements  to  be  re-corrected 
from  more  extended  scries  of  observations,  all  the  details  of  this  first  approxima- 
tion will  be  omitted.  The  resulting  elements  of  Uranus  are  given  in  the  follow- 
ing table,  togctlicr  with  the  adopted  elements  of  Saturn,  which  are  nearly  the  same 
as  those  employtd  in  tlie  theory  of  Neptune,  except  that  the  inclination  and  lon- 
gitude of  the  node  have  been  corrected  to  agree  witli  observations: — 


Elements  II.  of  Urnnus. 

Elrnionts  I.  of  Saturn 

m 

168°     16'     31" 

90°       4'       0" 

$'-'' 

28      25      36.0 

14      48     45.0 

¥ 

73      11      58 

112      20       0 

f 

0      4(5      20 

2      29     39.2 

c 

.0469276 

.0560050 

e  in  seconds, 

9679.5 

11551.9 

n 

15426.10 

43996.13 

m 

1 

21000 

1 

3501.6 

In  romputing  the  perturbations  cf  tiio  radius  vector,  one  of  the  largest  torms 
will  be  a  constant.  To  avoid  the  necessity  of  computing  separately  the  perturba- 
tions of  the  second  order,  which  depend  on  this  constant,  we  shall  include  an 
approximate  value  of  it  in  the  mean  distance.  This  approximate  value  is,  in  the 
action  of  an  outer  or  an  inner  planet,  h\o^a  =  —  \  m'Ma^D^  h'f.  In  the  action  of 
an  inner  or  an  outer  planet,  Mog  a'  =  +  J  mM(b"l^  -f  a  Dab'f).  M  being  the 
modulus  of  the  system  of  logarithms. 

Using  the  values  of  hf  and  a  7).  h'l\  which  ore  found  in  different  works  relating 
to  Celestial  Mechanics,  wo  find  that  the  different  planets  produce  the  following 
changes  in  6  log  a,  the  units  being  those  of  the  seventh  place  of  decimals: — 


V 


32 


TUE    ORBIT   OF   URANUS. 


On  Saturn. 

On  Uranns 

Action  of  Venus, 

+   22 

+  22 

Earth, 

+  24 

+  25 

"        Mars, 

+    3 

+     3 

"         Jupiter, 

+10865 

+8780 

"         Saturn, 

+3081 

"         Uranus, 

—       35 

"         Neptune, 

—         9 

—   119 

"         Sum, 

+ 

.0010870 

+ 

.0011792 

S  log  a 

+ 

.0001812 

+ 

.0001965 

The  uncorrected  mean  distance  is  deduced  from  the  mean  motion  by  the  rela- 
tion 

We  thus  have 


Saturn. 

Franns. 

Uncorrected  mean  dist.  (log) 

0.979496 

1.282901 

Action  of  the  planets 

+     181 

+     197 

Corrected  log  a 

0.979677 

1.283098 

The  following  functions  of  the  elements  are  derived  from  the  preceding  ele- 
ments by  well  known  formuhc: — 

y  (mutual  inclination)  1°  5  '  24.4:" 

log  sin  i  y  ==  (T  8.232373 

log  cos  1  y  9.9999307 
T  (long,  of  ascending  node  of  Saturn  on  Uranus)  126°  44'  51" 

o  41    31    40 

w'  323    18    21 

<y  — 6)  =  (u)  281    46    41 

log  sin  (u)  —9.990759 

log  cos  (u)  +9.309888 

fin2((,))  —0.39966 

cos  2(g))  —0.91667 

a  0.497249 
1 


a 


4.04438 


The  following  functions  of  a,  necessary  in  computing  the  coefficients  ?i,  are 
derived  from  llunkle's  Tables,  published  by  the  Smithsonian  Institution: — 


«^A 


THE   ORBIT   OF   URANUS. 


Values  of  a^Dlb'lK 


i 

6<« 

aDab'» 

a'Dlb'l' 

a'/>V/J 

a«i):6| 

a'Z)'6'J 

0 

2.14447 

0.33969 

0.5878 

1.081 

3.44 

13.6 

1 

0.55207 

.68314 

.4990 

i.in 

3.40 

13.8 

2 

0.20836 

.47198 

.7396 

1.152 

3.59 

13.9 

3 

0.08687 

.28491 

.7123 

1.463 

3.68 

14.5 

4 

0.03793 

.16270 

.5632 

1.596 

4.30 

15.1 

5 

0.01702 

.09010 

.3998 

1.485 

4.87 

16.9 

6 

0.00777 

.04896 

.2653 

1.231 

4.98 

19.1 

7 

0.00359 

.02624 

.1682 

0.940 

4.60 

20.5 

8 

0.00168 

.01392 

.1022 

0.675 

3.91 

20.4 

9 

0.00079 

0.00733 

0.0615 

0.463 

3.11 

18.8 

Dciivadrcs  with  re-fjtcct  to  (log 

'({  =  »)  o/a 

»/>:?/;>. 

i              n 

=    0* 

1 

^2 

3 

4 

DJ>*>                 IM 

aDah^f)         Dx 

(oi^Z)^/,';')    D. 

WI^'J.'I')      A,(a*Z);i'» 

0            0.33969             0.9275 

2.257 

6.68 

27.4 

1 

68314             1.1821 

2.175 

6.93 

27.4 

2 

47198             1. 

2116 

2.631 

7.05 

28.3 

3 

28191             0.9972 

2.888 

8.07 

29.2 

* 

16270             0. 

7259 

2.722 

9.09 

32.3 

$ 

09010             0.4899 

2.285 

9.33 

36.4 

6 

04896             0.3143 

1.762 

8.67 

39.0 

7 

02624            0.1944 

1.276 

7.4i 

38.9 

1 

01392             0.1161 

0.879 

5.93 

36.0 

ft           0.00733             0.0688 

0.586 

4.50 

31.2 

& 

coiid  (Jcrivativcs. 

f 

Dii'? 

Z>;(a2)o6"?) 

Diwi^f'V 

2)-(a'^;6'|>) 

© 

0.9275 

3.184 

11.19 

47.4 

:t     . 

1.1821 

3.357 

11.28 

48.2 

t 

1.2116 

3.843 

12.31 

49.4 

$ 

0.9972 

3.885 

13.85 

53.4 

,* 

0.7259 

3.448 

14.53 

59.6 

t 

0.4899 

2.775 

13.90 

64.4 

t 

0.3143 

2.076 

12.19 

65.0 

Vo 

lues  of  a"+^Dlb"> 

« 

^ 

a'Djr^ 

.Wl'''^ 

«'/'i6| 

$ 

1.865 

2.674 

8.104 

30.8 

^     1, 

1.267 

2.844 

7.77 

30.8 

f 

0.761 

2.412 

7.63 

29.9 

.            t'^ 

0.433 

1.790 

6.92 

28.7 

.       t 

0.240 

1.224 

5.73 

26.8 

ft 

0.130 

0.792 

441 

23.5 

g^ 

0.070 

0.493 

3.2U 

19.6 

k 

April.  1873. 

8a 


"f- 


34 


THE   ORBIT   OF   URANUS. 


Derivatives  with  rcs2>ect  to  {log  a  =  ».) 


1 

V„{a'Dabp 
=fi<»,7i<i> 

0 

4.539 

13.452 

65.1 

1 

4.111 

13.46 

54.1 

2 

3.173 

12.45 

52.8 

3 

2.223 

10.50 

49.5 

4 

1.464 

8.18 

44.0 

5 

0.922 

5.99 

36.7 

6 

0.563 

4.19 

29.2 

Second  derieatii'ca. 

i 

a^DvB^» 

a,DvH<! 

ia^DvBV 

0 

4.539 

17.99 

82.0 

1 

4.111 

17.f.7 

81.0 

2 

3.173 

15.6:^ 

77.7 

3 

2.223 

12.72 

70.5 

4 

1.464 

9.64 

60.4 

5 

0.922 

6.91 

48.7 

6 

0.563 

4.75 

37.6 

a,E^'> 

a 

,i5<? 

ia,E<-i 

Gfl,£<f 

a,Dr,E<'>         a,DvE'2 

UJKE'V 

0 

1.267 

4.111 

13.46 

54.1 

4.111         17.57 

81.0 

1 

1.313 

3.856 

12.95 

54.0 

3.856         16.80 

79.8 

2 

0.850 

3.167 

11.98 

51.8 

3.167         15.15 

75.8 

3 

0.500 

2. 

318 

10.31 

48.4 

2.0I8         12.63 

69.0 

4 

0.281 

1. 

573 

8.24 

43.1 

1.573           9.82 

59.6 

5 

0.155 

1.014 

6.18 

36.6 

1.014           7.20 

49.0 

The  notation  B'l^  and  Ej;  is  that  of  liC  Vcrvier  in  liis  devolopment  in  the  first 
volume  of  '■^Anwdea  de  V Ohwrvnioirc  Inq)erii.d  dv  Paris." 

Numeric d  exjircssion  of  R  and  its  dciivatircs. 

We  next  proceed  to  the  computation  of  the  coefficients  h  and  their  derivatives. 
As  an  example  of  the  most  convenient  form  of  computation  we  present  in  full  that 

of  the  coefficient  of  —  cos  (jX' — (t— 1)X — w)  in  the  expression  of  R  for  the  action 

"1 
of  Saturn  on  Uranus.     In  this  computation  I  use  the  tahles  given  by  Le  Verrier 

in  his  "Annales  de  VOhsermtoire"  tome  i,  ])ajTes  358-383,  comparing  the  develop 

ment  with  that  of  Professor  Tcircc  in  the  Astronomical  Journal,  vol,  i,  as  a  control. 


(•aw 


THE   ORBIT   OP   URANUS. 


86 


y=o; 

y=i- 

• 

-3 

—2 

—1 

0 

+1 

2 

3 

4 

6 

6 

i> 

+4 

3 

2 

1 

0 

—1 

—2 

—3 

—4 

6 

(0)Xt<,'> 

+0.5212+0.8334]+  1-1041 

0 

—1. 10414 

—0.83344 

-0.6212 

—0.3034  — 0.1702'-0.0929J 

0)y.alJ^bf 

—0.2849 

—0.4720—  0.6831  —0.3397 

—0.68314 

—0.47198 

-0.2849 

—0.1627 

—0.0901  -0.049(1 

Ao,  (50)(0 
a,  (50)(«) 

-16.1775 





+O.2303 

+0.3614—15.7565 

—0.3397 

—0.78728 

—1.30542 

—0.8061 

—0.4601 

—0.2603 

—0.1419 

(0)6«j" 

(i)x«/>- 

—13.55 

—11.25    —     5.52 

0.00 

0.00 

+  2.92 

+  5.73 

+  6.83 

+  6.46 

+  8.54 

+  7.7S 

+    4.78 

+0.51 

0.00 

+  1.18 

+  2.56 

+  3.25 

+  3.06 

(2)X«««. 

+  1.43 

+  0.74 

—     0.00 

—0.59 

—  1.00 

—  2.22 

—  2.65 

—  2,82 

—  2.40 

(3)X«»fi'« 

—  0.73 

—  0.5S 

—    0.59 

—0.54 

—  0.59 

—  0.68 

—  0.73 

—  0.80 

—  0.74 

Aa,(51) 
(1,(51) 

+  48.53 



+  4.71 

—  4.31 

—  3.31 

+  47.20 

—0  62 

—  1.69 

+  1.30 

+  6.46 

+  6.38 

(0)X6'/' 
mX'D,. 

—18.70 

—13.33 

—    4.42 

0.00 

+  4.42 

+13.33 

+18.76 

+  19.41      +17.01 

+  13.1o|  +10.38;  +     4.10'  —0.68 

—  1.37 

+  2.83 

+  6.27 

+  7.48      +  7.02 

(2)X.'/>'- 

+  1.43 

o.oo;  —  1.00  —2.35 

—  3.00 

—  6.92 

—  7.13 

—  6.70 

—  5.60 

(3)X«'i^« 

—  1.46 

—  1.15    —     1.18    —1.08 

—  1.18 

—  1.15 

—  1.46 

—  1.60 

—  1.48 

ao,(D2) 
1,(52) 

+  32.36      

—  5.69    —  4.10    +  29.86'  —4.11 

1                1 

—  1.13 

+  9.09 

+16.44 

+  18.53 

+16.96 

(0)X"^,(') 

—3.00 

—  3.40    —    2.63        0.00 

+  2.63 

+  3  40 

+  3.00 

+  2.25 

+  1.55 

(1)X<',£,(') 

+2.32 

+  3.17    +    3.8(1    +4.11 

+  3.86 

+  3.17 

+  2.32 

+  1.57 

+  1.01 

Aii,(tiO) 
0,(60) 

+  16.18      

+  17.41    +4.11 



+  6.67 

—0.68 

—  0.23 

+  6.49 

+  6.32 

+  3.82 

+  2.60 

1<X(50) 

+6.611 

+  10.12,  -441.22    -9.51 

—50.049 

1 

-36.556 

-22.57:1 

—13.05 

—  7.28    —3.97 

J,'X(51) 

—0.09 

—  0.07    +     1.04    —0.01 

—  0.035    +  0.028+  0.103 

+  0.14 

+  0.14 

Jee"x(52) 

—0.08 

—  O.oo'  +     0.45    —0.06 

i  —  0  017    +  0.138+  0.24S 

+  0.28 

+  0.26 

iec«X(tiO) 
A 

—0.05 
+6.39 

—  0.02 
+  9.97 

+     0.14    +0.03 

+  0.051    +  0.052+  0.043 

'                  i 

+  0.03 

+  0.02 

'  —439.59    —9  55 

—50.050    —36.338-22  179 

—12.00 

—  6.8u|  —3.60 

The  derivatives  of  h  with  respect  to  (log.  a  =  r))  are  computed  in  precisely  the 
same  way  by  simply  substituting  for  h[\  aDab'\\  etc.,  their  derivatives  with  respect 
to  »  as  given  in  the  above  table  of  constants. 

The  quantities  Ai,(50)"\  etc.,  which  appear  in  the  third  series  of  terms  above 
express  that  part  of  the  perturbations  of  Uranus  caused  by  the  action  of  Saturn 


III  units  of  tlic  tliird  pluro  of  dpoimiils. 


36 


THE    ORBIT   OF   URANUS. 


on  the  sun.  They  arc  each  of  the  form  .Y  X  a~^  N  being  a  numerical  coefficient 
given  by  Le  Vorrier  under  tlio  coefficient  for  each  term.  The  derivative  of  tliis 
expression  with  respect  to  »  is  —  2N  X  ct~S  so  that  for  the  corresponding  terms 
in  DJi  and  Dih  we  have 

ADJi  =  —  2Ah 

ADlh  =  +  iSh 

The  values  of  h  and  its  derivatives,  corresponding  to  any  one  argument  f  and  i, 
are  to  be  combined  into  two  terms  depending  the  one  on  the  cosine,  the  other  on 
the  sine  of  the  argument.  Let  us  represent  by  g  the  mean  anomaly  of  Uranus, 
and  let  us  put  T  for  the  mean  longitiide  of  Saturn  counted  from  the  perihelion  of 
Uranus,  or,  more  exactly,  for  the  arc  X — w.     Put  also 

Then,  for  each  value  of  N  there  will  be  several  values  of  P  corresponding  to  dif- 
ferent powers  and  products  of  the  eccentricities  and  inclinations  in  h.  Distin- 
guishing these  values  and  tlie  corresponding  values  of  h  by  subscript  numerals,  we 
shall  have  a  series  of  terms  of  B  of  the  following  form — 


22  =  ^ 


7i.  cos  (.V+  P,) 
-{-h,cos{N-{-P'^) 
+  7»3Cos(iV+P'3) 
-(-  etc.  etc. 


and  by  putting 


T'l- 


h^  =      7<,  COS  P\  -(-  hi  cos  P'„  -(-  7/3  cos  P^  -\-  etc. 
h,  =  —  7*1  sin  P\  —  7*2  sin  P.,  —  hj,  sin  P\  —  uic. 

iibove  terms  may  bo  condensed  into 


(41) 


72  =      7j,,  cos  iV  -f"       '*»  s^"  ■^' 


which  are  of  the  form  supposed  in  the  preceding  theory. 

In  order  that  the  derivative  of  R,  with  respect  to  the  true  longitude  of  Uranus, 
may  be  expressed  in  the  form 


fiR       m       .     „  ,    m  ,- 

„    =  —  r,  sm  iV  +       k.cosjY 

&V  M,  f», 


we  must,  by  (7),  put 

v.  =  — (i+j\)  hi  cos  P\ 
'»c=— (i+j\)  hi  sin  P\ 


■  (i  -\-jt)  hi  cos  P\  —  etc. 
(t  -j- ji)  '*2  sin  P2  —  etc. 


(42) 


j\>  j-2i  representing  the  several  values  of  j  in  the  different  terms  which  correspond 
to  one  and  the  same  set  of  values  of  i  and  t". 


jl 


THE   ORIJIT   OF   UK  ANUS 


87 


To  obtain  the  dcrivativo  with  respect  to  y  wo  notice  that  all  tlie  apprcciahlc 
terms  in  the  difleicnt  values  of  /<,  whicli  depend  upon  the  mutual  inclination,  are 

of  the  form 

li  =  ff-  A, 

where  <t  =  sin  J  y.     These  equations  give 

v7i      oh  da        ,  ,  1    J    ■ 

Aa cos  ^y  =  ^  A  sui y. 


Consequently 


6y      da  vy 

OR 

iJy 


l^lsinycos(.V-|-/'). 


and  the  various  terms  depending  on  the  same  argument  («',  /)  may  be  condensed 
into  two,  exactly  as  in  the  case  of  It  itself. 

Tlic  different  co-efficients  U  and  DJi,  computed  in  the  way  already  described,  are 
given  in  exlcnso  in  the  following  table.  At  the  top  of  each  individual  column  is  given 
the  value  of  P,  or  of  jij  -\-j\'i\  corresponding  to  the  values  of  h  below,  and  imme- 
diately under  P  is  given  its  modified  value,  or  P',  to  be  used  in  condensing  the 
terms,  putting  for  brevity 

((.))  =  (J  —  (j'. 

P  and  P'  are  therefore  regarded  as  constant  angles  the  numerical  values  of  the 
sines  and  cosines  of  which  may  be  obtained  from  the  values  of  u  and  cj'  already 
given. 

The  condensed  /t„  and  h,  are  given  in  the  two  right  hand  columns. 

All  the  numbers  are  given  in  units  of  the  third  place  of  decimals. 


Values  of  h.                                                                1 

r 

0 

w   U 

p 

=  0 

(") 

h. 

h. 

i      i' 

h 

h 

—!.  +  ■* 

4-  0.02 

—3,     3 

+  0.82 

—2,     2 

4-  0.88 

—1,     1 

4-  0.53 

0,     0 

4-10'?2.90 

—  0.4975 

4-  1072.80 

+  1—1 

— 34Hl.3r. 

—  0.83 

—  34S1.42 

—    1.33 

2^ 2 

+  205.02 

—  9.(13 

4-    204.14 

—  10.29 

3,-3 

+     84.27 

+  l.(!2 

+       84.77 

+    0.79 

4,-4 

■j-     35.  SO 

4-  1.54 

+      3(i.24 

+    0.90 

5,-5 

+     15.48 

+  1.2 

4-      15.81 

+    0.81 

/' 

10 

/ 

U 

u— 2u' 

UJ  -  -  2w 

1 

u 

i" 

0 

-(-) 

—  2(u>) 

+  (") 

u-j-w 

+2m> 

K 

h. 

i      i' 

h 

/, 

h 

/( 

/. 

—3, +  4 

—     2.31 

4-     f..39 

0 

0 

0 

0 

—     1.01 

—     0.20 

—2,     3 

—     3.42 

4-     !).!t7 

—0.95 

0 

0 

+  .02 

—     0.51 

—     9.40 

—1.     2 

—     3.54 

—439.59 

0 

0 

—.03 

+  .02 

—  93.30 

+  430.35 

0,     1 

+287.30 

—     9., 55 

+0.13 

—  .02 

—.04 

4-. 09 

+  285.20 

+     9.19 

+  1-     0 

+  58.38 

—  50.05 

—0.02 

—  .05 

—.03 

0 

+  48.16 

+  48.90 

2,-1 

—  3!).7fiO 

—  3(!.:W8 

—0.074 

—  .071 

—.016 

0 

—  47.138 

+  35.531 

3—2 

4-  35.13 

—  2-2. 179 

—0.105 

—  .313 

—.007 

0 

+  30.f,3 

+  21.44 

4,-3 

4-   20.3!) 

—  12.(!0 

—0, 1 1 

+  -n 

0 

0 

+   17.90 

+   12.55 

ft,— 4 

4-   11.25 

—     CMC, 

—0.10 

4-   .23 

0 

0 

+     9.98 

+     7.00 

r.,— 5 

4-    ('..00 

—     3.f. 

0 

+   .22 

0 

0 

+     5.38 

+     3.78 

88 


THE   ORBIT   OP   URANUS. 


Valuks  of  h. 

P 

—  2u 

—  W    —  U 

—  2«' 

0 

P" 

0 

h 

-2(«) 

+  2« 

h. 

/,. 

i      i' 

h 

h 

h 

—1,     3 

+0.04 

—  0.27 

42.24 

+0.11 

+38.72 

+  17.03 

0,     2 

+0.00 

+31.05 

—  0.04 

+0.18 

+  0.58 

—31.15 

+  1.     1 

—0.74 

—  0.83 

—  1.08 

—0.01 

—  0.03 

+  2.15 

2,     0 

+3.00 

—  5.19 

+   1.89 

+0.18 

+  0.32 

+  4.14 

3,-1 

+  1.2(i8 

—  0.200 

+  2.082 

+0.111 

—  1.912 

+  5.216 

4,-2 

+4.1(i 

—  5.458 

+   1.707 

+0.001 

+   1.44 

+  4.570 

5,-3 

+3.13 

—  4.03 

+   1.20 

+0.04 

+   1.13 

+  3.30 

(1,-4 

+2.11 

—  2.74 

+  0.87 

+0.02 

+  0.75 

+  2.31 

7.-5 

+  1.35 

—  1.70 

+ 

0.55 

+0.01 

+  0.49      1 

+   1.40 

P 

—  3« 

—  u  —  2u 

.Jui    u 

—  3„' 

t 



Id 

P' 

0 

-(-) 

—  2M 

—  :!(w) 

t)U u 

2« 

h. 

h. 

i      i 

h 

h 

h 

/(, 

/) 

h 

+  1.+2 

0 

—.12 

0 

—.11 

—.14 

+.02 

+.17 

+  .00 

2,     1 

+.01 

—.07 

+  .03 

—.02 

—.02 

+  .01 

0 

+  .04 

3,     0 

+.18 

-.40 

+  .34 

—.08 

—.02  . 

+  .04 

— 15 

+  .213 

4,-1 

+  .222 

—.700 

+.520 

—.117 

—.010 

+  .020 

—.325 

+  .410 

5,-2 

+.424 

-.847 

+.,558 

—.122 

—.012 

+  .021 

—.170 

+  .400 

(!,-3 

+.:i08 

—.773 

+  .408 

—.107 

—.008 

+  .014 

—.145 

+  .400 

:,— 4 

+  .327 

—.027 

+  .305 

—.080 

—.005 

+.009 

—.107 

+  .370 

8,—;-. 

_-t-Al  __: 

—.47 

+.20 

—.06 

0 

0 

—.083 

+  .280 

A 

—  4w         1  —  u' Itio 

—  2«'  —  2u 

ow  u 

—  ij 

P 

0          i      —  («) 

—  2(o) 

—  3(..) 

—  4(") 

;.. 

/'. 

i     i' 

h 

h 

h 

h 

h 

4,     0 

0 

—.04 

+.04 

—.02 

0 

—.04 

0 

5,-1 

+.02 

—.08 

+.08 

—.04 

+  .01 

—.03 

+  003 

<■),— 2 

+  .030 

—.105 

+  .100 

—.047 

+  .008 

—.045 

+  .028 

7—3 

+  .043 

—.114 

+  .112 

—.048 

+  .008 

—.049 

+  .031 

«,— 4 

+  .042 

—.100 

+  .103 

—.044 

+  .007      1 

—.044 

+  .032 

»,— 5 

+.0;$.') 

—.000 

+  .080 

—.030 

+.000 

—.037 

+  .029 

Vai.les  of  DJi. 

P 

0 

lo    b) 

F 

0 

(«) 

i    i 

D„h 

/>../l 

Dvho 

D«h, 

-4. +4 

+    3.18 

—3,     3 

+    3.24 

—2,     2 

+    2.39 

—1,     1 

+    0.40 

0,     0 

+  171.03 

—    2.075 

+    171.51 

+  1—1 

+8740.50 

—    2.09 

+  8749.12 

—    3.02 

2, 2 

+  407.72 

+  20.83 

+    472.40 

+  18.05 

3,-3 

+  277.00 

+     2.87 

+    278.25 

—    0.36 

4,-4 

+   153.87 

+    4.02 

+     155.46 

+     1.41 

5,-5 

+     82.12 

+     4.00 

+       83,54 

+     1.80 

THE   OK  HIT   OP   URANUS. 


30 


*- 

Vai.i  Ka  (IF  7Vi.                                                                       1 

F 

—  u 

t 
—  W 

u— 2u' 

«'  — 2- 

+u' 

u 

F 

0 

-(") 

—  2(«) 

+  (") 

Dvh 

2w 

Dvh, 

Dvh. 

i     %• 

Vvh 

Dvh 

Dvli 

Dvh 

Dvh 

—3, +4 

—     9.28 

+   19.4 

0 

0 

0 

0 

—     5.32 

—  19.00 

-2,     3 

—    9.74 

+   18.5 

+  2.05 

—  .06 

—.12 

+.08 

—     7.97 

—  17.43 

-1,     3 

—     4.fi3 

+907.62 

+   .05 

—  .08 

—.13 

+  .07 

+  1K0.44 

—888.65 

— ",     1 

— 5.5(i.40 

—  26.17 

—  .10 

—  .09 

—  14 

—.12 

—561.82 

+  25.60 

1,     0 

+  30.16 

—  71.56 

+   .04 

—  .19 

—.11 

+.02 

+  15..37 

+  69.80 

2,-1 

+  2(1,-).  17 

—  86.47 

—  .19 

—  .26 

—.07 

0 

+247.58 

+  84., 30 

3,-2 

+  83.10 

—  74.60 

—  .42 

+  .63 

—.04 

0 

+  68.34 

+  73.49 

4,-3 

+  68.34 

—  54.9 

—  .,55 

+   .29 

—.02 

0 

+  5V.70 

+  53.9 

6,     4 

+  49.04 

—  ;!6.9 

—  .58 

+  .68 

—.01 

0 

+  42.19 

+  36.7 

(>,— 5 

+  32.1 

—  23.2 

_  .53 

+  .90 

0 

0 

+  28.1 

+  23.4 

P 

—  2u                —  w  —  w 

—  2w' 

0 

F 

0 

-(") 
Dvh 

_2M 

+  2« 

Dvh, 

Dvh. 

i     { 

i>o/l 

Dvh 

Dvh 

-l,+3 

+  0.16 

—  0.66 

+  86.02 

+0.47 

— 7S.76 

—34.19 

0,     2 

+  0.32 

—64.55 

+  0.15 

+0.60 

—  12.92 

+  (;2.54 

+  1.     1 

+  3.27 

—  2.70 

+  4.69 

+3.02 

—  1.22 

—  2.20 

2,     0 

+  2.,')6 

—  8.78 

+  5.00 

+0.60 

—  3.74 

+  6.00 

3,     1 

4-13.745 

—15.870 

+  7.232 

+0.463 

+  3.933 

+  12.153 

4,-2 

-flO.2,') 

—IS.  93 

+  7.85 

-1  0.32 

—  0.77 

+  15.07 

,5,_3 

+  10.76 

—  IS. 07 

+  6.99 

+0.21 

+  0.69 

+  14.68 

(1,-4 

+  ',1.39 

—  15.0 

+  5.59 

+0.13 

+   1.22 

+  12.4 

T,— 5 

+  7.3 

—11.5 

+  4.21 

+0.09 

+   1.2 

+  9.0 

P 

—  3« 

_ 

_2«  — u' 

,  — «  — 2w' 

•1  ' 

—  u        — u 

P 

i    i 

0 

—  (") 

1     —  2(") 

-3(„) 

^u  —  u'    +2u 

Dvh', 

Dvh. 

Dvh 

Doh 

Dvh 

Dvh 

Dvh 

1   Dvh 

+  1.  +  2 

+0.03 

+0.18 

+  0.05 

+0.22 

+  .22 

1  +.08 

—0.31 

—0.17 

2,     I 

+  0.12 

—0.26 

+o.:i3 

— 0.09 

—.08 

+  .17 

—0.08 

—0.09 

3,     0 

+0.21 

— 0.S9 

+  0.98 

—0.31 

—.09 

+  .13 

—0.59 

+0.13 

4,-1 

+0.S9 

—2, ((2 

+  1.S9 

—0.54 

—.09 

+  .13 

—0.84 

+  0.69 

;'>,— 2 

+  1.0S 

—3.00 

+2.53 

—0.67 

-.08 

+  .11 

—1.38 

+  1.30 

(•),— 3 

+  1.40 

—3.47 

+2.73 

—0.69 

—.07     +.09 

—  1.33 

+  1.68 

1,-4 

+  1.46 

—3.44 

+  2.60 

—0.(12 

,  —.05 

+  .06 

—1.20 

+  1.78 

8,— >) 

+  1.32 

—3.00 

+  2.20 

—0.51 

1       0 

!  +.04 

—0.99 

+  1.62 

P 

—  4« 

■ ,iui W 

—  2a— 2« 

—  u  —  ow 

-w    1 

P 

i      i' 

0 

—  (u.)                —  2(u,) 

_3(„) 
Dvh 

4(«) 

Dvh, 

Dvh. 

Dvh 

Dvh               Dvh 

1 

■)vh 

4,     0 

+  .02 

—.09      '      +.13 

-.08 

+  .02 

—.06 

—.01 

fi,— 1 

+  .0.5 

—.22      ,       +.30 

—.18 

+  .04 

—.14 

—.02 

(;,— 2 

+  .10 

—.38       '       +.48 

—.26 

+  .05 

—.24 

+  .01 

■J,— 3 

+  .15 

—.52             +.62 

—.32 

+  .00 

—.30 

+  .04 

«,    -t 

+  .18 

—.59             +.66 

—.33 

+  .06 

—.32 

+  .09 

n,— 5 

+  .19 

— .59              +.65 

—.31 

+  .05 

—.31 

+  .11 

The  values  of  I)vfi,  needed  in  compntins;  the  perturbations  of  the  second  order 
with  respect  to  the  masses  being  obtained  in  the  same  way,  by  the  simple  substitu- 
tion of  the  second  derivatives  of  the  functions  h,aDuh,  etc.,  for  those  functions 
themselves  in  the  expressions  for  /*,  it  is  not  necessary  to  present  the  details  of  the 
computation. 

After  obtaining  h  and  its  derivatives.,  ic  will  be  found  convenient  to  change  the 
arrangement  of  tlie  terms.  Ilitlurto  we  have  kept  in  one  series  those  in  which 
the  sum  of  the  indices  are  a  constant.     Now,  we  shall  put  together  all  those  in  which 


40 


THE  OKBIT   OP  UIIANUS. 


he  index  of  the  disturbing  planet  has  the  same  vnhie,  nrranfriug  the  individual 
terms  ot  each  scries  according  to  the  index  of  the  disturbed  phmet.  Thus  the 
index  of  the  product  of  any  term,  as  h  cos  N,  by  any  multiple  of  the  mean  anomaly 
of  tlic  disturbed  planet,  as^y,  "ill  be  found  in  the  same  series  witli  that  of  .V  itself 
and  J  lines  above  and  below.  ' 

The  next  process  will  be  the  formations  of  the  required  functions  of  the  mean 
anomaly  of  Uranus,  |^J,  '^J^,  "', ,  bg  r.     Their  values  are  as  follows  :- 


rr=  = 


Po 


=  » 


=  1 


1.001  l();i  +.0()().55()7  "'''  "'^'~ 

+.()9;3!);J3  ros    g  -.04(i8SS«)  cos    g  +.0468889  sin    g  +.0938294  cos    g 

Iniw.    '"'!^  -.001(i494  cos -V  +.0032988  sin  2y  +.0055012  cos  2^ 

tZtt  '"  ?  -;;;|"<)^^-^  -«  3y  +.0002190  sin  3^  +.0003357  cos  3^ 

+.000020  cos  4y  -.0000035  cos  Ag  +.0000142  sin  4g  +.000020G  cos  Ag 

Considering  only  those  terms  wuich  are  of  the  first  order,  the  value  of  D'^R  may 
be  found  in  two  >yays,  tlie  agreement  of  whicli  will  aft'ord  a  check  upon  the  entire 
dcvvelopment  of  the  perturbative  function,  and  upon  the  computations  of  7?  and 
These  are  (1)  by  direct  differentiation,  witli  respect  to  the  time  as  con. 


ov 


tamed  in  the  mean  anomaly  of  a  single  planet,  whereby  each  term  in  R  of  the 


will  produce  in  D\R  tlie  term 


R  =■      /*  cos  N 
a. 


D\R  =  —  "*  inh  siniV 


and  (2)  by  forming  the  expression 


D\R  =  "'^  '•'■» 
6'v    (It 


GR  dp, 
"^  Op    lit  ' 


As  several  "mechanical  multiplications,"  like  those  indicated  in  tliis  hst 
expression,  are  to  be  performed,  the  following  example  of  the  fom  o^  com- 
putation IS  presente.1.     It  exhibits  the  formation  of  the  product  of  tliose  terms 


of  -.:    in  wliich  t'  =  _  1  by 


ih 


-  '  -^^  (sin) 
m   S" 

X  .04G915 


X  .002750 


-'^          -10                 +1                  2  3               4  .5 

+  .02     +   l.fi.'i     _2S7.42       +:!4,S1.14  +  r,4.42  +  C.988  +1.18  +0.10 

I       "             0          -i-        OH      —     l;!.48  +10,1.32  +  2  5r)3  4-0  ■?<?  -Lft  nr 

U.08     -13.48     +.r.3...2       +       ,,„  I     0  33  +  ^J^^  +1''  +;o« 

'       °             "                  ^                     0«<'  -     0.79  +  9.570  +0.15  +    02 

+          02              0  0                0  0 
"                   0  —     .048  +  .58  0 


i-_.79  +  9.58  +     0.15 

X  .0001G8       +.68  +  0.01  0 

a,    f)R   di^  ,  .   ^      T,  '7                                                              — ____ 

m-n-di  dl    ^''""^—^^  -^'^^  -^-•^■^'^       +3170.23      +217.28     +19.125     +2.24     +0.18 


>f^i^t»ism''i"iimii&imm 


THE   ORBIT   OF   URANUS. 


41 


The  Tinltiplicr^  on  the  left  arc  each  one-half  the  cocfflcinnt  cos  j<j  in  the  ex. 

prcssion  for    ,." ,  and  cacli   product  is  phiced  in  the  two  columns  corresponding 

respectively  to  N-\-j<j  and  N — jy. 

All  the  derivations  of  /?„  necessary  in  the  computation  of  the  perturbations  of 
the  first  order  are  given  in  the  following  tables.  First  we  have  the  values  of 
D\R  obtained  by  direct  differentiation,  as  indicated  in  the  preceding  formidic. 

,  obtained  by  the  formulir  (7)  and  (42).     The  products 


Next  we  have 


dv 


and 


f)E 


6. 


"'"  and  of  ^^'  by  ^^,  being  formed  in  the  simple  way  just  pointed  out, 


6R^ 

and  with  the  values  of'the  component  factors  just  given,  their  sum  is  next  shown, 
This  sum  should  agree  accurately  with  D',l{.  The  discrepancies  are  shown  in  the 
next  two  columns.  The  only  apparently  large  discrepancy  is  found  in  the  argu- 
ment Otf' — 5/.     It  probably  arises  from  the  incompleteness  of  the  computation  of  R 

and  -      ,   so  far  as  they  depend  on  this  argument.     As  the  entire  term  docs  not 

amount  to  0".01,  I  have  not  sought  to  correct  it. 

The  great  value  of  this  check  arises  from  the  fact  that  it  gives  a  complete  con- 
trol of  the  correctness  of  the  development  of  the  perturbativc  function,  ab  initio, 
since  the  two  valves  of  D',R  are  derived  from  diff'erent  terms  of  that  development. 

dR 

It  also  controls  all  the  computations  except  that  of        .      This  quantity  being 

op 

multiplied  by  quantities  of  the  order  of  the  eccentricities  in  the  second  value  of 

D'lR,  an  error  in  its  value  will  produce  a  discrepancy  of  only  ■^\f  its  own  amount  in 

jy,R,  and  niny  therefore  be  overlooked.     The  derivative  in  question  must  thertv 

fore  be  checked  by  a  complete  duplicate  computation. 

In  the  column  next  following  are  given  the  integrating  factors  r,  for  which  the 

expression  is 

n  1 

V  = 


i'u'-\-  i)i 


i  +  i' 


For  each  value  of  /'  the  values  of  r  arc  therefore  the  reciprocals  of  a  scries  of  num- 
bers iu  arithmetical  progression,  the  common  ditfcrcuce  being  unity. 


April,  1873. 


49 


TIIK   OIiniT   OF   URANUS. 


'^' 

/V«  = 

m' 
-     n  X 

<5v  ~ 

-  X 

«1 

(lU       in' 

'J     V 

Hi  1 

I'oa 

Kill 

C(I9 

cos 

ttiri 

0,     0 

0 

0 

. .  •  • 

4-     0.4S7 

—1 

244.31 

.... 

'i 

— 

4S.15 

4-  48. 9C 

-f      10.20 

4-   49.07 

— 

03.52 

—118.82 

3, 

— 

O.fU 

-I-     8.28 

-f-       4.48 

4-     3.20 

+ 

3.42 

—   10.14 

3, 

+ 

0.45 

+     0.(!4 

+       0.50 

—     0.08 

+ 

0.74 

4-     0.34 

4, 

+ 

O.IC 

0 

-f-       0.00 

—     0.00 

+ 

0.10 

4-     0.01 

—  2,-1 

0 

+     0.08 

4-       0.02 

—     0  04 

■+ 

0,08 

—     0.05 

—  1, 

— 

0.03 

+     2.15 

+       1.03 

4-     1.34 

L 

1.25 

—     0.11 

0, 

0 

0 

—  287.42 

—     0.10 

+ 

176.02 

4-  34.70 

1, 

f3ISl.42 

—     1.33 

+  3481.14 

—     1.04 

— .'J 

2r7.70 

4-    4.;i5 

2, 

+ 

94.28 

f  71.00 

+     54.42 

4-  71.23 



200.44 

—  119. .S3 

•'!, 

■V 

.'■..74 

+   15.05 

-1-       0.988 

+     9.490 



2.02 

—  17.37 

4, 

+ 

1.30 

+    i.(;o 

-f        l.lH 

4      0.42 

4- 

1.10 

—     1.11 

\ 

+ 

0.15 

-f     0.15 

-1-       0.10 

0 

+ 

0.17 

—     0.01 

-1,-2 

-f 

0.17 

+     O.Ofi 

4-       0.21 

—   0.10 

+ 

0.14 

—    o.u 

0. 

0 

0 

—       ^.58 

+   30. 119 

+ 

6.34 

+   31.39 

1. 

+ 

93.30 

— 4;!0.35 

4-     90.81 

— 4;i0.3,• 

— 

87.14 

—458.30 

2, 

— 

40M.2S 

_  20.58 

—  410.42 

—  12.01 

— 

070.00 

—     7.70 

•■?. 

— 

91. HO 

+  04.32 

—     57.01 

—  04.98 

— 

98.97 

—  94.93 

4, 

— 

r..7G 

-f   IS. 30 

4-       1.45 

4-   12.90 

— 

0.07 

—   19.05 

5, 

+ 

0.S8 

+     2.45 

4-       1.30 

4-     1.04 

4- 

1.50 

—     1.79 

«, 

+ 

0.27 

+     0.17 

4-       0.20 

—     0.02 

+ 

0.28 

—     0.04 

1,-3 



38.72 

—  17.03 

—     38.75 

—  17.29 

4- 

40.04 

—  17.16 

2, 

+ 

1.02 

+   IS. SO 

i        5.31 

4-   19.14 

+ 

8.48 

—  20.83 

3, 

— 

254.31 

+     2.37 

—  254.15 

—     0.02 

— 

303.02 

—     0.43 

4, 

— 

71.84 

+  50.20 

_     61.49 

+  49.90 

— 

75.00 

—  00.45 

5, 

— 

5.()D 

+   10.95 

—       0.23 

4-   13.0 

— 

1.82 

—  18.07 

fi, 

+ 

0.8V 

-;       2.76 

4-       1.29 

4-    1.5 

+ 

1.48 

—     2.14 

7, 

+ 

0.34 

-)-     0.24 

4-       0.27 

0 

4- 

0.35 

—    0.07 

3,-4 

+ 

3.03 

+   J8.78 

4-       1.34 

4-   18.78 

4- 

6.33 

—  25.26 

4, 

— 

144.90 

4-     3.00 

—  144.78 

+     1.5 

— 

191.70 

—     2.31 

s. 

— 

49.00 

+  .35.00 

—     38.04 

+  34.58 

— 

52.17 

—  43.7 

fi, 

— 

4.50 

-f   13.80 

—       0.9 

4-  11.2 

— 

1.97 

—  14.7 

7, 

+ 

0.75 

+     2.05 

4-      1.1 

4-     1.0 

+ 

1.31 

—    2.2 

8, 

+ 

0..35 

+     0.20 

4-       0.3 

+    0.1 

-f 

0.30 

—    0.1 

5,-5 



79.05 

+     4.05 

—     78.9 

4-    2.4 

— 

99.3 

—     2.1 

c, 

— 

32.3 

+  22.7 

—     20.3 

4-  22.3 

— 

33.5 

—  27.2 

7. 

— 

3.4 

+   10.4 

—       1.1 

+     8.7 

— 

1.7 

—  11.1 

8. 

+ 

0.7 

+     2.3 

4-       0.9 

4-    1.5 

+ 

1.1 

—     1.0 

9, 

+ 

0.3 

+     0.3 

1 

4-       0.3 

4-    01 

+ 

0.3 

4-     0.1 

w 


T  II  K   O  II  HI  T   O  V   U  U  A  N  U  8. 


43 


w 


111' 

olt    </,.„ 

oV     <" 

DiMtTf 

mncy. 

V 

<■. 

<•. 

a    I' 

=  «,"X 



Hill 

t'()9 

Hill 

CU» 

0,     0 

0 

—     0.01 

0 

—.01 

.  .  .  • 

—1244.31 

1, 

—     4H.0') 

-{■  48.92 

+  .10 

—.04 

+  1.0 

+     32.78 

—  20.90 

2, 

_       0.03 

+     8.27 

+  .01 

—.01 

+  0.5 

+       4.00 

—     1.80 

.% 

+       0.4,') 

+     0.05 

0 

+  .01 

+  i 

+       0.44 

+      O.OH 

4, 

-1-       0.11 

—     0.03 

—.05 

—.03 

+  0.25 

_2,— I 

—       0.03 

-f     0.02 

-.03 

—.00 

-0.200008 

+       0.08 

—     0.08 

—  1, 

—       0.05 

+     2.10 

—.02 

+  .01 

—0.259001 

+       1.23 

—     1.23 

0, 

—       0.00 

+     0.09 

—.00 

+  .09 

—0.350023 

+  270.02 

+  34.79 

+  1 

-f34Sl.41 

—     1.33 

—.01 

0 

—0.539942 

—1508.18 

+     5.78 

2 

+     94.20 

+  71.00 

—.08 

0 

—1.173030 

+     20.85 

—280.59 

ii. 

+       .1.78 

+   15.03 

+  .04 

—.02 

+0.75940 

—     79.50 

-}-194.17 

4, 

+       1.27 

+     1.00 

—.03 

0 

+0.871120 

—       1.10 

+     1.79 

T', 

-1-       0.14 

-(-     0.11 

—.01 

—.04 

+  0.4050 

+       0.03 

+     0.13 

\ 2 

+       0.10 

+     0.10 

—.01 

+  .01 

—0.149102 

+       0.19 

—     0.13 

0, 

+       O.Ol 

+     0.07 

+  .01 

+  .07 

—0.175312 

+       0.34 

■1    31.39 

+1. 

+     93.24 

430.37 

—.00 

—.02 

—0.212580 

—     47.47 

—275.33 

0 

—  40S.2S 

—  20.00 

0 

— .(i-J 

—  0,209971 

—  897.05 

+     3.35 

3, 

—     91.90 

+  04.33 

—.01 

+  .01 

—0.309800 

—  100.93 

—142.50 

4, 

—       5.T7 

+   18.17 

—.01 

—.13 

—0.580813 

—       7.44 

—  41.13 

5, 

+       0.91 

+     2.45 

+  .03 

0 

—  1.42022 

+       4.05 

—     8.75 

c, 

+       0.29 

+     0.15 

+  .02 

—.02 

+  3.3797 

—       1.53 

+     1.09 

1,-3 

—     38.78 

—  17.03 

—.00 

0 

— 0.1,32342 

+     29.73 

—  12.05 

2, 

+       1.02 

+   18.08 

0 

—12 

—0.152528 

+       8.79 

—  32.50 

3, 

—  254.29 

-f    2.:i7 

+.02 

0 

—0.179981 

—  451.55 

—     1.28 

4, 

—     71.88 

-1-  50.13 

—.04 

—.07 

—0.219482 

—  107.20 

—  88., 50 

r', 

—       5.09 

-f   17.01 

—.04 

+  .00 

—0.281202 

—       5.00 

—  27.01 

G. 

+       0.91 

+     2.73 

+  .04 

—.03 

—0.391210 

+       2.15 

—     4.30 

7, 

+       0.35 

+     0.19 

+.01 

—.05 

—0.0426 

+       0.79 

—     0.38 

3,— 4 

+       3.10 

+  18.78 

+  .07 

0 

+  0.11893 

+       7.05 

—  29.73 

4, 

—   145.0 

+     3.51 

—.04 

-.09 

—0.13500 

—  230.83 

—     3.28 

5, 

—     49.9 

+  34.85 

0 

-.15 

—0.15005 

—     07.74 

—  54.00 

fi, 

—      4.7 

+   13.95 

—.20 

+  .09 

—0.18490 

—       3.03 

—  19.80 

t, 

-'r       0.8 

+     2.02 

+  .05 

—.03 

—0.22085 

+       1.05 

—     3.30 

8, 

+       0.3 

+     0.2 

—.05 

—.00 

—0.2934 

+       0.57 

—     0.27 

5,-5 

—    79.3 

+     2.7 

—.25 

—1.35 

—0.1080 

—  110.4 

—     3.5 

fi, 

—     32.4 

+  22.7 

—.10 

0 

—0.1211 

—     41.3 

—  32.7 

7, 

—       3.5 

+  10.5 

—.10 

+  .10 

—0.1377 

—       2.6 

—  13.9 

8. 

-(-       0.8 

+     2.3 

+  .10 

0 

—0.1597 

+       1.3 

—     2.0 

fl. 

+       0.3 

+     0.2 

0 

—.10 

—0.1901 

+       0.5 

—     0.2 

u 


THE   on  BIT   OF    URANUS. 


R' 


The  values  of  "'  f  D',Ii„ilt  rire  formed  from      ,'  ITEhy  simple  multiplication  by 
V,  and  proper  changes  of  sign.     The  vuUies  of  k^  and  Ic,  arc  then  formed  by  adding 

the  terras  of  2   ' .'  (  i>«iio'''  to  the  corresponding  terms  of      ', 
111 "'  "« 


in'    d^ 


Perturhalions  of  radiu6  vector. 
Let  us  now  resume  equation  (19),  and  put  for  brevity 


(44) 


■«.(l  +  »0 

If  we  give  to  u  the  successive  values  0,  +1,  —1,  -f2,  —2,  -fS,  —3,  we  have 

+  i S< (iVA(-i)  +i>u-i)V() ('—''(i-o)  1  ''".cos (.V+  g)  -\-k. sin(iV+i/) | 
r,=,^p=i/'X  ]  +|2.(i>,<z^,+„+i'(.+i)'7.X'*— »'-(M-i))l^"oCos(^—  sr)+/.-,sin(JV^— «7); 

-|-  etc.  etc.  "tc.  etc. 

the  finite  integral  being  taken  with  respect  to  all  values  of  i  from  —  oc  to  +  oc, 
and  the  terms  in  which  the  angles  A'iwf/  vanish  being  omitted.  Piocceding  far-, 
thcr  to  expand  with  respect  to  /,  if  we  collect  similar  terms  we  shall  find  the  indi' 
vidual  terms  in  j-'.^p  to  be  as  follows: 

j  Pill  (»'i  —  '-i) 


•^  i  +   M'j  ('•»  — ''-0 

Ml   ("i  —  »'") 
txnr)  +  (P'l^  +  P'h)  {»'a  —  ''-0 
+  *^^  ^  +(M3  +  M«)('V-*-.) 
-f-  etc.  etc. 


4-JJ^ 


»V/i    ('o  —  I'-i) 

-f  (Mi+M^)  (''2  —  ''-:0 

-|-  etc.  etc. 


>   I  A'o  cos  iV  -|-  /i",  sin  iV  ( 

.  \hcQs{N+g)+k.smiN+g)\ 

■  ;  Z-„  cos  (iV -  flf)  +  A-.  sin  (JNT -  i/)  ( 


•f  i  ^  •  +  (Pili  +  Pill)  (''3  —  v-i) 

I  4-  etc.  etc. 


1 


{l;cos{N+2g)-\-l:s{n(N-\-2g)l 


+  \M 


+  i^r 


{2Wi  (»'j  — J'-j)    ■) 
+  ( Piqs  +i'3?.)  (»^,  -  »'_3)     \  J /.•„ cos  ( A^-  2g)  +  Z-,  sin  (AT -  2g)  \ 
4-         etc.  etc.  ) 


+ 

p»q3  (»'»  — n) 
+  (pi74+r.?i)(»'«  — »'-i) 

+  (7'i72— i'//i)(»'2  — J-i) 
-|-  etc.  etc. 


j  /.-.  cos  (iV+  3sr)  + 1;  sin  (JV  +  3g)  \ 


m 


TUE   ORBIT   OP  UBANUS. 


i6 


+  i^ 


+  (/"74+ivy.)(".  -»■-)!  I /,„ cos (iyr- 3^7;  +  /.-. sin (iyr_3j7) I 

+  {Pi'h  —  iV/i)  ("-1  —  »'-2)  I   ' 
-|-  etc,  etc.         I 

A  law  of  the  factois  of  /.-,  cos  (iV+  «//)  +  A-.  sin  (^+  ««/)  wliicli  will  be  noticed 
in  the  above  expression,  is  this:  lloprcsenting  this  i:tctor  by  A'„,  wo  have 

the  index  i  representing  the  coefficient  of  (j  in  iV,  so  that  only  half  the  valnes  of 
it' need  be  separately  computed. 

As  the  computation  of  r,«<^p  from  these  formulie  .an  be  arranged  in  such  a  way 
as  to  be  very  simple,  the  computation  of  the  termv  i.i  which  the  index  i'  is  —1  is 
Here  presented  quite  fully.  The  logarithms  only  are  omitted,  being  used  only  in 
the  cases  in  which  they  are  more  convenient  than  a  table  of  products.  In  prac- 
tice I  find  it  convenient  to  write  them  in  red  ink  immediately  under  the  numbers 
which  tliey  represent. 


via 


First,  to  find  M,  it  will  be  noticed  that  in  the  expression       .,"V'     y  the  a  in  the 

a,(l  -j-  VI) 

numerator  represents  the  moan  <Hstance  of  the <//«/iuie7 planet,  as  deduced  from  the 

observed  mean  motion  by  the  equntion  («'«'  -:^u  (1  -|  "0  while  «,  represents  the  mean 

motion  of  the  onler  planet.     When  the  outer  planet  if  the  disturbed  one,  the  ratio 

"-  would  be  unity,  but  that,  to  avoid  r    large  class  of  second  or.'-r  terms,  o,  has 

been  corrected  for  perturbations  in  the  beginning  (p.  32).     In  the  case  of  Uranus 
disturbed  by  Saturn,  we  have  in  consequence 

a 


Whence 


log       =  ;).<)91)8()3. 


i/=  285.44 


in  iniits  of  the  sixth  place  of  decimals. 

Computing  the  values  of />(  and  q^  from  (U?)'  we  find,  for  Uranus, 


i  M  mi  —  — 

•i  -iVC/v/i— 7'//i) — I- 


0.0784 
10,044 
3.343;> 
0.0028 
0.2;5.')8 
O.llh 
0.008 
0.005 
0.()();{ 


In  units  of  the  sixth 
place  of  decinuds. 


In  the  computation  the  first  three  lines  are  copied  from  previous  pages. 


46 


THE   OllBIT   OF    URAxNUS. 


i  =  —  1 

• 

-4 

—3 

—2 

—1 

0 

+1 

2 

3 

4 

5 

V 

-0.1460 

-0.1700 

-o.soeio 

-0.35000 

-0.85062 

-0.68004 
-1608.18 

-1.17868 

+6.76040 

+0.87118 
—1.10 

1- 0.466a 
40.03 

k„ 

+0.08 

+  1.38 

+276.62 

+  20.85 

-  70.56 

+142.56 

•l— •-! 

-0.08 

-1.28 

+  84.70 

—  .28034 

+  6.7U 

-286.60 

+  104.17 

+2.04476 

+  1.70 

—6.2938 

+  0.18 

—0.6536 

—  .0887 

—  .14452 

—  .82301 

+7.299:14 

+     0.0784 

.,-._, 

—  .2047 

—  .3690 

—  .9675 

+7.0190 

+1.2218 

+  1.0O85 

+  1.4913 

— 6.51b6 

—  10.044 

'i~'o 

—  .0.'),'?5 

—  .0910 

—  .1893 

—  .63361 

+7.9330 

—5.8883 

—0.4056 

+     3.343:t 

'i—'l 

—  .1797 

—  .3338 

—  .9140 

+  7.1100 

+1.4110 

+  1.6392 

—6.4417 

-«-     0.0028 

'..-». 

—  .39 

—1.00 

+6.96 

+  1.13 

+0.82 

+0.86 

+  1.41 

—    0.23B8 

»»—  •  V 

.144 

—  .280 

—  .823 

+7.299 

+2.045 

—6.29  J 

+     0.118 

•a— '-1 

—  .3(;9 

—  .9i;8 

+7.019 

+  1.222 

+  1.005 

+  1.-190 

0.008 

—  .33 

—  .91 

+  1.41 

+11.82 
+7.93 

+1.64 
+O.Mi 
—5.89 

+     0.005 
+     0.0003 

1.00 

+6.97 
—  .19 

—20.60 

•j-'-l 

—  .09 

142.56 

X{'|-'-,) 

—12.35 

— 39.9(;5 

—117.327 

+  1040.59 

+291.49 

—897.2 

—78.8 

0.0784 

X(',— -,) 

—  0.02 

—  0.03 

—  0.075 

+0.550 

+0.10 

+0.08 

+0.1 

—  05 

—    10044 

A'o 

-12.87 

-80.68 

+0.914 

-40.010 

+  1.901 

-110.777 

+  1040.00 

+201.57 

+B0.I4 

-807.1 

-70.8 

X  (•,-•„) 

+0..'i32 

+  6.364 

—79.68 

+  4.1 

+     3.:tJ:l3 

Xf»i-»-,) 

— O.OCl 

-1.12ii 

—3.056 

+23.770 

+  4.72 

+  6.48 

-21.B 

+    o.ot>a: 

X(«j-»-j) 

—0.001 

-0,003 

+  0.020 

+  0.003 

0 

0 

0 

A', 

—  0.20 

—1.186 

+80.187 
-  1.135 

—1.721 

— 74.»« 

+64.62 

—17.4 

—  .2:i.')a 

A-:. 

Xfvj-O 

—0.07 

+o.oi;« 

—0.205 

+  0  194 

+80.14 

—0.48 

-74.1I6 

+  1.48 

+64.6 

+0.034 

+   .118 

X('j-»-,) 
A'. 

—0.04;; 

—0.114 

+0.!-2S 

+  1.022 
-0.00!' 

+0.144 
—  1.577 

+0.1-2 
-0.86 

+  0.18 

+  1.66 
+  1.58 

.008 

A-*. 

-0.048 

+  1.03 

—.013 

+  .003 
.(H)5 

X  (•,-•„) 

X  {•,-»,) 

+  .007 
+  .035 

-.011 

+  .005 

+  .004 

-.007 
-.003 

+   176117 

+.004 

^.000 
+.(M2 

+  21698 

h'„  h 

—  1 

—  2il 

—11076 

—23197 

+  991 

2 

K,    h 

0 

0 

0 

—314 

-45152 

—15113 

—51  •■•. 

+  19 

K  ,  h 

0 

—  5i! 

+  1712 

-t-628 

+  5964 

-71 

0 

0 

Kj   /■, 

0 

0 

n 

0 

+   2,-2 

-1-2378 

—8 

-132 

—  3 

+  72 

+  21 

+  126 

0 

0 

0 

0 

K'k, 

0 

0 

0 

0 

0 

0 

+  8 

0 

A',/-„ 

+  3 

+   1 

0 

0 

0 

0 

0 

0 

r,i»,(.Oi.) 

-  1 

-    0 

—0848 

f 170557 

-17508      -82458 

-4151 

-116 

X.0-.     •• 

i        0 
0 

0 

0 

-43S 

-l->2.-3            —-21 

— lii.',;t 

-1!)5 

—438 

+  8283 

—821 

—1053            —195 

—  5 

X. 002751 

—26 

+:!(> 

0 

+485 

0 

-48 

0 
-62 

—26            +485 
—  11                     0 

— 4H 
II 

—62 
0 

X.OOOIfix 

—     3 

-  4 

—  1 

0               —2 

-4-30 

—  3 

cos  ^1^  (co.-») 

+  8 

+  84 

+  26 

-1112 

-1393 

+  175235 

—676 

-10315     -22086 

— 2982.'i4     +56614 

-5227 

— 1606 

-875 

—  10 

+  1 

A,    <. 

0 

0 

0 

—  39 

+  175  !  +21483 

+  12548 

—31 

A'  ,t. 

0 

—    7 

—6 

— »(;37 

—14556          +116 

-2 

0 

A,  i. 

0 

0 

0 

0 

+  36 

—9 

+  104 

+  322 

(1 

n 

—293 

—306 

—     1 

0 

0 

0 

A'f,i, 

0 

—  11 

II 

I) 

0 

0 

0 

-  I 

r,«»,  («in) 

+  1 

+    8 

-1602 

-0058 

-812600 

J  1*8204 

+  11044 

4  281 

X. 0409 15 

{        0 

0 

—  79 

0 
—453 

—  79 
— Ilt.ilii 

-453 

+3i;(i'.i 

-146116 
+  518 

+  3lil>!l 
+  13 

+  51H 
II 

0 

0 

0 

—  5 

-27 

—860 

+  215 

X.  002751 

—  27 

— HIIO 

+  215 

+  3M 

0 

0 

0 

x.oooiim 

.^ 

—  53 

4-    13 

-1-2 

0 

0 

'1 

-  3 

COs*l^(:.ill) 

-  » 

-  151 

-  21tlt3 

341M6 

-800351) 

4  64020 

f 18864 

4!>«1 

it" 


•  . 


THE   O  K  B 1 T   OF    URANUS. 


47 


' 


In  forming  the  next  ten  lines,  it  will  be  noticed  that  the  value  of  r„  correspond- 
ing to  any  vertical  column  is  found  u  columns  to  the  right.  It  is  therefore 
necessary  to  extend  the  line  v  two  columns  at  cath  end.  The  extension  on  the 
right  is,  however,  omitted  for  want  of  space.  In  performing  the  subtractions  it 
will  be  convenient  to  copy  the  j^'s  again  on  the  lower  edge  of  a  horizontal  strip  of 
paper,  and,  in  forming  tlie  differences  r„ — r„'  to  lay  the;  strip  above  the  line  of  r's, 
and  u — u'  colunnis  to  the  riglit. 

On  the  left  of  each  line  of  differences  is  written  the  factor  by  which  that  line  is 
to  be  multiplied. 

The  mode  of  formation  of  the  A"s  is  evident  from  the  formula. 

It  will  be  seen  that  the  same  computation  which  gives  7C  gives  also  K_„,  only 
the  latter  belongs  u  columns  to  the  right. 

Each  l\  and  //,  is  multiplied  in  succession  by  all  the  A"s  wliich  lie  below  it  in 
the  same  cohuun,  but  the  product  by  A',  is  to  be  written  u  columns  to  the  right, 
and  that  by  A'_„  u  columns  to  the  left.     The  sum  of  the  products  in  any  one 

colinnn  gives  the  coefficient  of  ^?^  {Ig  -\-i'l')  in  the  development  of  r,*!^, 

</i\         »«°  cos  4/         ,  ,5  ...         , 

This  quantity  being  multiplied  by    ^n~  =      ^.z      we  have  cos  4!'p,  which  only 

needs  to  be  inultii)liod  by  sec  v^  =  l.OOllO:}  to  give  5p.     The  unito  of  ri\^p  and 
^p  correspond  to  the  ninth  place  of  decimals. 

All  the  periodic  terms  are  to  be  treated  in  this  manner,  all  the  series  of  values 
of  l-^  and  /•„  inchuling  the  constant  term,  being  subjected  to  the  same  process. 
Hut,  when  /'  and  /  are  both  zero,  v  will  be  infinite.  Here  we  sinii)ly  omit  the  r, 
treating  it  as  if  it  were  zero.  We  thus  obtain  the  coni[)lete  value  of  the  terms 
with  constant  coefficients  in  ;V-'\)  and  >^)  wliich  are  given  in  the  following  table. 
Tlie  terms  midtiplied  by  the  time  are  still  to  be  comptited.  They  are  derived  from 
('20),  wliich  may  be  put  in  the  form 

Thisexp^ession  is  computed  thus: 


u 

Pn 

*<;" 

«7.. 

*•;'        /'.,/.•'"' 

-qJ-V 

0 

—  .t4l)-s;j 

-  1  •.'44.31 

0 

0       +n->.i8 

0 

1 

4-.!)!)!! '7 

+     ;5-.Ms 

4-  .!»!»«)■ 

•i 

— 'io.oo       +  ;}J.7:) 

-f  2(1  >i) 

2 

+  .()t>:U 

-t-        4.()G 

+  smi 

—   1.8G         +     ().(»!) 

+  0.04 

We  have  now 

^ /'„/.•':;'  = 

-  _|_  OOS.Ct?; 

\  }r:iii..i-":^  =  +  -.'nosn 

-:iqj-r  = 

:.+    '.'().!»:} ; 

iJ/'iv,./.- -+  -mn 

r^\> 

=  —  'iU)  lit 

-\-  '.'!)«()  lit 

cos      If 

+ 

'>{)(»S1  lit  sin    g 

4-  ■<<»  III 

COS  '2l/ 

+ 

()!)7  ///  sin  'i'/ 

+          -',7 

COS  % 

+ 

'2i  it  sill  ;{</, 

in  units  of  the  ninth  ])laee  of  decimals.     The  value  of  cos  ^i.^p  is  obtained  from 
them  by  multiplying  by  /[',  exactly  as  in  the  case  of  the  constant  terms. 


48 

THE  ORBIT  OF    URAXUS. 

n««p 

cos 

4«p 

log  My 

V. 

V. 

9     V 

cos        1 

sin 

COS 

sill 

0,     0 

—210  ni 

—70  nt 

+  139)i7 

1, 

-I-29HG  H< 

+29081  nl 

+2H7«n< 

+29033  nt 

2, 

-1-70  Hi 

+097  nl 

+  209  »i< 

+  2084  >i< 

3, 

-f  2  »< 

+  24  H< 

+  14/1/ 

+  139«< 

0,     0 

—355045 

-354347 

oc 

0 

1, 

-I-14S75 

—1503 

—18411 

—1496 

2.45551 

—2912 

+14206 

2, 

— 2,S3 

+  49 

—1530 

—21 

2.25448 

—039 

+457 

3. 

—20 

—  1 

—112 

2 

1.97839 

—40 

—8 

—2,-1 

—  1 

+   1 

+  3 

—  9 

1.7090 

+  1 

+3 

-1, 

—10 

+  8 

+  34 

—151 

1.8098 

+  12 

—99 

0, 

—9343 

—1093 

—1112 

2!t92 

2.00035 

—2870 

+  10 

+  1, 

-|-17(i557 

—9058 

+  175?35 

—21186 

2.18780 

+530512 

+  100 

2, 

—17508 

—312600 

—10315 

—309359 

2.52504 

+  1823 

—23800 

3, 

—22453 

+78204 

-22980 

+04029 

3.28542 

—13482 

+  18310 

4, 

—4151 

+  11044 

—5227 

+  13804 

2.39559 

—293 

+  104 

5, 

—115 

+281 

—373 

+  901 

2.1235 

—13 

0 

-1-2 

—  1 

+   1 

+   1 

2 

1.0292 

+  9 

+1 

0, 

—5(5 

— 27G 

+32 

—  100 

1.0993 

—329 

—1.551 

+1. 

+710 

+3723 

+  1075 

+  3728 

1.78303 

+.5874 

+  20114 

2, 

+20180 

—13 

+  20583 

+  484 

1.88083 

—31000 

+  920 

3, 

+7797 

+0512 

+  8824 

-1-0796 

2.02348 

—0019 

—6800 

4, 

+  1534 

+0202 

+  2054 

+  0230 

2.22401 

+243 

—2172 

5, 

+  2071 

—5954 

+  2134 

—5010 

2.00780 

+527 

—421 

0, 

—053 

+755 

—550 

+  494 

2.98438 

—190 

—20 

1-3 

—151 

+  04 

—137 

+  75 

1.5772 

—1464 

+  053 

2, 

— i-)2 

+  221 

+  149 

4  229 

1.03880 

+  231 

—833 

3, 

+4348 

+  15 

+  4420 

+88 

1.71074 

—13050 

4' 

4, 

+  1504 

+  1280 

+  1774 

+  1314 

1.79091 

—3225 

—3126 

5, 

+  133 

-{-085 

+  214 

+756 

1.9045 

—18 

— 1')43 

«, 

—111 

+  232 

—109 

+273 

2.0479 

+  144 

—  103 

7. 

—103 

+  87 

—108 

+  100 

2.2034 

+  50 

—7 

3,-4 

—28 

+  121 

+31 

+  122 

1.5408 

+  181 

—038 

4, 

+  1223 

+  n 

+  1245 

S         +41 

1.5858 

—.5578 

— f.8 

5, 

+  485 

+  389 

+  545 

1        4  400 

1.6488 

—1721 

—15(0 

<■', 

+38 

!       +201 

+03 

+  221 

1.7225 

—50 

—590 

7, 

—25 

+53 

—23 

+  03 

1.8012 

+70 

—100 

S. 

—15 

+  7 

—10 

+  9 

1.9229 

ft,—:) 

+393 

+  12 

+  400 

+  18 

1.4889 

—2432 

—74 

fi. 

+  175 

+  138 

4  194 

+  142 

1.5385 

—90S 

'       —700 

7, 

+  14 

+77 

-+22 

!         4  84 

1.5940 

—40 

— .UO 

8, 

—10 

+20 

—  9 

4  2» 

1,05M<) 

1         +40 

—70 

9, 

—  5 

+  3 

—  5 

4    4 

1.7:!45 

^ims^ 


I  ■itr-Bir  r*— "*"•""  wnaniriiriinMiM 


THE   ORBIT   OF  URAXUS. 


49 


Pert  111  hations  of  Lomjitude. 

The  perturbations  of  the  longitude  are  now  to  be  computed  by  formulte  (24). 
To  do  this  iu  the  most  simple  way  we  remark  that  the  numbers  given  on  page  42, 

under  the  heading  '^'^  are  those  represented  in   formula  (42)  by  v,  and  v,.      If 
c*v 

we  put 


nt  =  i 


equation  (24)  may  be  put  into  the  form 


di 


out  we  have  from  (42) 

„    ill  =  S(        V,  sin  N—        V,  cos  N). 
If  now  we  represent  the  numerical  values  of  cos  i|-(^p,  already  found,  by 

2  (p,  sin  .V  -f  p„  cos  N) , 
and  if  we  substitute  these  expressions  in  the  above  value  of     . ,  ,  the   latter  will 


become 


d'v 


'y^,  =  rr- S  Kv.  -  2l'.)  si"  ^V+  (V,.  -  2p,)  cos  N\, 

where  we  put  for  brevity 

V,  =      Mn;., 

The  nnmoriral  expression  for  r,""  is  given  on  page  40,  and  by  multiplying  the 
quantities  witliin  brackets  by  tiiis  expression,  after  tlie  manner  explained  on  pages 

40  and  41,  we  form  the  terms  of  '  ,'f.     Multiplying  each  of  these   terms   by  its 

al 

corresponding  value  of  r,  changing  ros  to  s!n  and  sin  to  ros,  we  have  the  coefficients 

iu  the  expressions  for  I'v  given  on  page  50. 

Al  previously  mentioned,  before  commencing  the  above  computation,  I  had 
conii.-''ed  all  the  perturbations  of  Uranus  by  the  method  of  "])erturbations  of  the 
elements,"  using  tlie  formuhr  developc^d  in  my  Investigation  of  the  Orliit  of  Ne|)- 
tune.  The  two  results  are  here  placed  side  by  sid(>,  for  the  purpose  of  comparison. 
The  discrepancies  in  the  various  coefficients,  expressed  in  thousandths  of  a  second, 
are  shown  iu  the  sixth  and  seunth  cohnuns. 

It  will  be  seen  that  the  largest  discrepancies,  and  indeed  the  only  ones  (with  a 
single  exception)  exceeding  one-tenth  of  a  second,  occur  in  the  coefficients  of  the 
terms  2/' — /  aug  %' — /.  Here  tiie  errors  are  almost  certainly  in  tlie  computation 
from  perturbations  of  tlie  elements.  Owing  to  the  limg  pi-riod  of  the  term  'i(/ — I 
they  would  not  become  sensible  iu  the  course  of  any  one  century. 

7         April.  1873. 


^ 


60 


THE   ORBIT   OF  URANUS. 


Pebtl'bbatiu.ns  or  the  Lonuitudk  of  Uranus  produckd  by 

TUB  ACTION  OF  SaTUBN,  AND           1 

UEI'ENUINU  UN  THE  I'lUST  I'UWEB  OC  TIIK  DISTL'BBINU  I'llUCKK. 

From  conip.  prucuiliiig 

From  purt.  of  uluiiiviits 
iu 

Discrepancy. 

0.434294  ip 

9,     I' 

8in 

coa 

siu 

cua 

.{ill 

cos 

cos 

8ia 

ti 

If 

fi 

If 

It 

II 

0,     0 

+  io.»(;90< 

+  10,9045/ 



.004.5/ 

1, 

—  1.2:10(1/ 

+   12.231  III 

—  1.22H  III 

+  12.271)1/ 

.002(1/ 

.040(1/ 

+  13(1/ 

+  128(1/ 

2. 

—  0.01-2  lit 

+  0.717  H< 

—0.072  nt 

+  0.720(1/ 

0 

.003  ((/ 

+  !((/ 

+  9(1/ 

3, 

—  O.OOiiil 

+  0043  «< 

—0.004/1/ 

+  0.043  ;i/ 

0 

0 

0 

0 

0,     0 

'"'*'* 

—1.541 

1, 

+  8.;")  45 

—     4.735 

+     2.844 

+     1.013 

.... 

.... 

—     80 

—  6 

2. 

+  0.4(>l 

—    0.1  ('.9 

+     0.133 

+     0.100 

.... 

—      7 

0 

3, 

-1-  0.028 

—    0.005 

+     0.013 

+     0.014 





0 

0 

—1,-1 

-f  o.o;;*; 

+     0.039 

+     0.032 

+     0.005 

4 

34 

0 

_       1 

0, 

+  1.282 

+     0.718 

+     1.280 

+     0.719 

2 

1 

—     5 

—     13 

1, 

—20.817 

+     8.522 

—  20.873 

+     8.595 

50 

73 

+  701 

—   100 

2, 

-11.890 

-1-143.4(13 

—  11.003 

+  143.405 

797 

2 

—  45 

—1.351 

3, 

+  4!t.30 

-|-115.8(i 

+  49.02 

+  110.08 

320 

220 

—103 

+  280 

4, 

+  2.1  13 

+     S.CIO 

+     2.195 

+     5.021 

02 

5 

—  24'+     01 

5, 

+  0.1 2(i 

f     0.329 

-j-     0.109 

+     0  331 

17 

2 

-     2  +       4 

0,-2 

+  0.017 

—     0.017 

+     0,025 

—     0.033 

8 

10 

0 

0 

1. 

+  0.042 

+     0.814 

+     0.034 

+     0.818 

8 

4 

+     7 

+     I"-. 

2, 

+  4.110 

—     0.009 

+     4,103 

—     0.012 

7 

3 

+  89 

+       2 

3, 

+  2.079 

—     l.(!07 

+     2.100 

—     1  070 

27 

G9 

+  38 

+     30 

4, 

+  0.048 

—     1.830 

+     0.043 

—     1.902 

5 

73 

+     9 

+     27 

5, 

+  1.1C3 

+     2.950 

+     1.274 

+     2.991 

111 

35 

+     9 

—     24 

6, 

+  0.503 

+     0,378 

4-     0,550 

+     0.445 

53 

07 

2 

+       2 

1,-3 

+  0.034 

+     0.012 

+     0,030 

+     0.015 

2 

3 

0 

0 

2, 

+  0.037 

—     0.041 

+     0,014 

—     0.050 

23 

9 

+     1 

+       1 

3, 

+  0.824 

—     0,019 

-1-      0,812 

—     0.017 

12 

2 

+   19 

0 

4, 

+  0.355 

—     0,207 

+     0,351 

—     0.203 

4 

4 

+     « 

+    fi 

5, 

+  0.047 

—     0.105 

-f     0,039 

—     0,191 

8 

20 

+     1 

+       3 

6, 

—  0.02r. 

—     0.003 

—     0,028 

—     0,000 

2 

3 

0 

+       1 

7, 

—  0.053 

—     0.032 

—     0.053 

—     0.018 

0 

14 

_     1 

0 

3,     4 

+  o.oor, 

—     0.022 

—     0,005 

—     0.023 

11 

1 

0 

+        1 

4, 

+  0,228 

—     0,008 

+     0,221 

+     0.002 

7 

10 

4-     5 

0 

5i 

+  0.103 

—     0.077 

+     0.084 

—     0,075 

19 

2 

+     2 

f       2 

fi, 

4-  0,013 

—     0,044 

1-     0,013 

—     0,057 

0 

13 

0 

+       1 

T, 

_  0.005 

—     0,013 

—     0.001 

—     0.015 

4 

2 

0 

0 

8, 

—  0.003 

—     0.002 

0 

+     0.002 

3 

4 

0 

0 

5,-5 

+  0.074 

—     0.003 

+     0.071 

0 

3 

3 

+     2 

0 

6, 

+  0.038 

—     0.027 

+     0.023 

—     0.020 

15 

1 

f      I'-f        « 

7. 

-f  0.005 

—     0.010 

-f     0,005 

—     0,025 

0 

ft 

0              0 

8, 

—  0.002 

—     0.004 

0 

—     0,003 

2 

1 

"1  » 

<4Ki* 


TUE   ORBIT   OF   URANUS. 


61 


cos  N 


Perturhationa  of  Latitude. 

These  are  computed  from  the  formuhc  (27)  and  (40),  no  reductions  being  made 
from  hk  and  l/i  to  hp  and  hq,  but  the  perturbations  of  the  hititude  being  computed 
directly  from  the  former  by  (40).  We  have  only  to  represent  the  expressions  for 
hk  and  hrihy 

hk  =  —  2«„  cos  N —  2a,  sin  N 
hr,  =      Sa',,  cos  iV-|-  2a',  sin  N 

and  substitute  u  for  n  in  the  equations  (40)  from  which  f)3  is  computed. 

The  principal  stops  of  the  computation  are  shown  quite  fully  in  the  following 
table.     The  values  of 

c7i       1        1      fJi 

are  first  formed  from  those  terms  of  Ii,  on  pages  37  and  38,  which  contain  d  as  a 
coeffici(!nt.     Then,  having  for  each  original  term  of  Ii 

OR  _  m'  eh 
dy       a,    dy 

all  the  terms  which  have  the  same  coefficients  of?,  and  ?.'  in  N  are  combined  into 
two  «lepoi!(ling  on  «/  and  /'  as  shown  in  the  case  of  Ii  on  page  3(5.  The  coefficients 
of  these  terms,  in  units  of  the  third  place  of  decimals,  are  given  in  the  columns 

licadcd  — . 

Oy 

The  value  of    '  sin  N  being  formed  for  each  term  of  R,  all  the  terms  depending 

on  the  same  multiples  of  ?.  and  ?.'  are  combined  into  two,  of  which  the  coefficients 
are  given  under  tlie  proper  heading.  Tiie  terms  of  (t  -\-j)ah  sin  N  being  formed 
in  like  manner,  we  have,  by  adding  the  last  two  expressions,  all  the  quantities 
which  enter  into  the  formulic  (27).  To  integrate  these  equations  thus  forming 
the  numerical  values  of  (V.-  and  hv;  we  have  only  to  multiply  each  term  in  the 
second,  third,  eighth,  and  ninth  columns  of  the  table  by  the  corresranding  values 

of    '" '"'  ,  for  which  we  juay  use  the  value  of    .   ''  „  already  given. 
<»,  cos  4/  sni  1 

The  quantities  giv(>n  in  the  four  columns  under  hk  and  —  hr,  show  the  values 

of  —  rt„  —  «,.,  —  («',,  —  rt'.,  corresponding  to  each  argument.      From   these    the 

terms  of  6;i  arc  formed  by  equation  (40)  with  the  modification  mentioned  above. 


52 


THE    ORBIT   OP   URANUS. 


PSRTIJllBATIONS  OF  TUB  LATITUDE  I'KODUCED  BY  SaTURN. 

3    '' 

dy 

2-'*-«iUiV 
'It 

I  ('+»'■'' sln^V 

:;;=.,/,.  X 

ik 

— i, 

W 

i      i' 

cos 

sin 

sin 

009 

sin 

cos 

sin 

00s 

sin 

It 

ens 

tt 

cos 

sin 

It 

Bin 

00s 

0,    0 

—10.82 

0 

0 

0 

0 

+0.008 

0.00 

+  .008 

+0.19S 

1, 

—  3.45 

—  2.92 

—1.94 

—  0.11 

+  0.17 

+0.84 

-  1.77 

+  0.73 

+0.203 

-0.172 

—0.104 

—0.043 

+0.246 

+0.209 

2. 

4-  1.42 

—11.10 

+  1.30 

+10.73 

+  0.07 

+0.05 

+  1.37 

+  10.78 

-0.042 

—0.326 

+0.042 

—0.317 

—0.049 

+0.0U 

3, 

4-  1.31 

-  1.71 

+1.31 

+  1.71 

+  0.01 

0.00 

+  1.32 

+  1.71 

—0.026 

—0.033 

+0.026  —0.033 

—0.008 

+0.003 

-2,-1 

+  1.0 

+  0.1 

—1.0 

—  0.1 

0.00 

0.00 

—  1.0 

—  0.1 

+0.012 

—0.001 

+0.012—0.001 

0.000 

0.000 

-1, 

—  e.43 

-52.« 

+6.4 

-82. 8 

+  0.03 

+0.02 

+  6.4 

—52.8 

—0  098+0.804 

—0.098—0.804 

—0.004 

+0.018 

0, 

—  0.97 

+  7.06 

+1.53 

+  5.14 

-  4.91 

0.00 

—  3.38 

+  S.14 

—0.144—0.145 

+0.070+0.106 

—0.661 

+0.678 

+1, 

+46.67 

0 

0.00 

0.00 

+59.45 

—0.02 

+59.45 

—  0.02 

+1.482     0 

—1.887—0.001 

+0.081 

—0.026 

2, 

—  1.84 

—  2.81! 

—0.99 

—  0.20 

+  0.93+1.32 

—  0.06 

+  1.02 

-0.127+0.197 

+0.004+0.070+2.218 

+1.92C 

3, 

+  0.90 

-  6.98 

+0.73 

+  0.44 

+  0.12+0.16 

1 

+  0.90 

+  6.60 

—0.358  —2  m 

1 

+0.358—2  630+0.091 

-0.071 

4. 

+  1.09 

—  1.38 

+1.09 

+  1.38 

+  0.02+0.01 

+  1.11 

+  1.39 

—0.056 

-0.071 

+0.057—0.071 

-0.048 

+0.068 

-1,-2 

+  7.6 

-  1.5 

—7.6 

+  1.5 

O.UO      0.00 

-  7.6 

+  1.5 

+0  067 

+"'"» 

+0.067  +0  013 

+0.(103 

+0.004 

0, 

+  13 

+10.9 

—1.3      +10.7 

—  0.11+0.53 

—  1.4 

+  11.2 

+0.013'— 0.112 

+0.014+0.116+0.008 

+0.034 

+1, 

—  0.27 

+  9.03 

+1.70    +  0.91 

+  1.85—7.35 

+  3.35 

—  6.44 

—0.003—0.112 

—0.042—0.080 

+0.040 

-0.04S 

2, 

-14.51 

0 

0.00 

o.do 

—  7.01  —0.21 

1 

-7.01 

—  0.21 

—0.230 

» 

+0.112—0.003 

1 

0 

—0.028 

3, 

—  2.85 

—  2.29 

—0.42    —  0.19 

-  0.97+1.11 

—  1.39 

+  0.92 

— 0.0()2+0.050 

+0.0,30+0.020 

—0.063 

-0.064 

4. 

+  0.43 

—  4.23 

+0.44    +  3.67 

+  0.02+0.22 

+  0.46 

+  3.80 

+0.015+0.146 

—0.016+0.134 

—0.080 

—0  014 

6, 

4-  0.80 

—  0.99 

+0.80 

+  0.9!) 

f  0.02+0.02 

+  0.82 

+  1.01 

+0.067 +i>  083 

-0.068+0.084 

+0.008 

—0.004 

2,-3 

—  1.17 

+  0.86 

+1.22 

+  0.97 

+  0.09+0.33 

+  1.31 

+  1.30 

—0.01(1          '0.* 

—0.011  +0.012 

+0.050 

—0.046 

3, 

—  8.54 

0 

0.00 

0.(10 

—  4.34'    0 

-..« 

0.00 

—0.090     0 

+0.046 

0 

+0.004 

-0.010 

4, 

—  2.08 

—  1.65 

—0.13 

—  0.15 

—  O.fS  +0.85 

-1.1 

+  0.70 

—0.026+0.022 

+0.013 

+0.009 

—0.012 

—0.015 

6. 

+  0.14 

—  2..'i0 

+0.24 

+  2.00 

0.00  +0.22 

+  0.24 

+  2.22 

+0.002+0.041 

—0.004 

+0.037 

—0.010 

-o.ooa 

«. 

+  0.55 

—  0.66 

+0.55 

+  0.6(! 

+  0.02+0.02 

+  0.57 

+  0.68 

+0.013+0.015 

—0.013+0.016 

+0.004 

—0.001 

3,-4 

—  0.72 

+  0.47 

+0.82 

+  0.78 

+  0.09  +0.,32 

+  0.91 

+  1.10 

_  0.005  —0.003 

1 

—0  006  +0.008 

+0.022 

—0.020 

4, 

—  4.80 

0 

0.00 

0.00 

—  2.4.S  —0.02 

—  2.4S 

—  0.02 

— 0.0:!8.     0 

1 

+0.020 

0 

+0.002 

—0.006 

5. 

-,.. 

-  1.09 

-0.01 

—  0.10 

—  0.06  +0.59 

—  0.6" 

+  0.49 

—0.013  +0.010 

+0.006 

+0.004 

—0  004 

—0.003 

6, 

+  0.12 

—  1.3 

+0.12 

+   1.0 

—  0.02+0.19 

+  o.in 

+   1.2 

+0.001  +0.014 

-0.001 

+0.013 

+0.0(>4 

+0.002 

5,-6 

—  2.64 

0 

0 

0 

—  1.35  +0.04 

—  l.,35 

+  0.04 

-0  017      0 

+  0.008 

0 

IXl L  A"  77  T 

S"^'"'"' ""•"•' {4  =  -4".  77  7- CO.  7. 

i 


^■■■I 


THE   ORBIT    OF   URANUS. 


53 


C  II  AFTER    III. 

PERTURBATIONS  PRODUCED  BY  NEPTUNE  AND  JUPITER. 

The  perturbations  of  Uranus  by  Neptune  were  originally  computed  with  ele- 
ments of  both  planets  quite  different  from  those  finally  adopted.  But  the  last 
computations,  on  which  the  concluded  values  of  the  perturbations  depend,  Averc 
made  with  the  concluded  elements  of  Neptune  found  in  my  investigation  of  the 
orbit  of  that  planet.'     They  are  as  follows : 


•  > 


7t, 

4;j 

17     .30 

0, 

l:j() 

7     33 

f. 

3;J5 

5     39 

«?>' 

1 

47       1.6 

«, 

7864.935 

e. 

().00849(J2 

log  a. 

l.-l 

78141 

Mass, 

TtJiTU 

rm» 


(J 

r 
(J 


Hence  follow  the  following  functions  of  the  elements  of  Neptune  and  Uranus: 

a  =    0.G38195 

12°     44'     58" 
247      45      20 
y  =      1      30     29.6 
c  =  sinjy=      0.013161 

M=  37.522  (in  units  of  6th  plito  of  decimals). 
From  these  values  of  the  elements  are  obtained  llie  following  values  of  the 
various  terms  in  the  development  of  the  perturbacive  function,  and  of  v.  As  the 
developments  have  been  formed  on  the  same  principle  as  in  the  case  of  Saturn,  it 
is  deemed  unnecessary  to  give  the  details  of  the  process.  It  is  only  necessary  to 
remark  that  the  indices  t'  and  i  arc  the  coefficients  of  l'  and  g  respectively,  the 
mean  longitude  of  Neptune,  or  I',  being  counted  from  the  perihelion  of  Uranus. 


'  Suiithsonian  Cuntributions  to  Knowledgi',  Vol.  XV. 


54 


THK   OR  HIT   OF   UIIANUS. 


A 

CTIIIN  or 

"'x 

Ni;i-n.Ni!. 

1 

/<- 

Ml' 

dlt 

'!"='"'  X 

«-r'>< 

V 

A. 

"1 

A. 

0'" 

"1 

(?P       "i 

aj 

•'     i 

*'» 

I'O 

00:1 

sill 

h 

d. 

0,      0 

+n:ir..(n 

0 

0 

4-  0.202 

4-3  ;8.2(i 

0 

4-.3(18.2(> 

0 

+  1 

— 

I».(i7 

—  I.2.'. 

f    1.01 

—  1.23 

—  (14.80 

—  5.89 

—  100  94 

—    8.39     4-  1  0 

+  2 

+ 

0.41 

—  0.0.1 

—    0.0 

—   0.1 

4-     2.98 

4-  0.12 

4-     3.82 

4-     O.Oli  '4-  0  5 

+  ■i 

U 

u 

0 

0 

0 

0 

0 

0          4-  0.33333 

1,-3 

+ 

0.21 

—  001 

4-    0.7 

—  0.2 

4-     ].(! 

4-  0.2 

+     2.1 

4-0.1       —  0.40158 

—  U 

— 

fi.21 

-  O.ii.1 

—   4.11  > 

4-  1.27 

—  211  5 

-f-  ((.(! 

—  43.5 

—     1.1     1  —  0.()7107 

—  1 

4- 

i;u.i!» 

+  (i.:ti» 

4-i:i3iis 

—  0.08 

4-509. m 

4-  0.77 

4-10.'-,7.49 

-f-     1.99     —  2.04023 

(> 

2.'). 4.') 

-1041 

—   18.0  1 

4-  0.02 

—  84  57 

—11.47 

—  84.57 

—  11.47     4-   1  9(11:17 

+  « 

+ 

1.12 

4-   1  19 

4-    0.5 

—   0.1 

4-    B.-.;i 

4-   0.82 

4-     7.31 

4-    1.07    4  o.««2;i2 

+  •■! 

0.05 

+   0.01 

4-    0.02 

+   0.02 

—    0.24 

0 

—     0.32 

4-     0.02  1  4-  0.39843 

2,-4 

+ 

O.S.I 

—  0.03 

4-    2.1 

0 

4-     2., 5 

4-  0.1 

4-     4.8 
--  59.7 

0 

—  0.335841 

—  3 

4- 

13.5:1 

—   1.01 

4-  27.7 

4-  3.1 

4-  18.7 

—  2.1 

—    5.5 

—  0.504971 

t) 

+ 

37.'..!I2 

+  0  :m 

4-751.59 

—  0.7! 

4-9  7.«2 

4-   1.25 

4-2491.(19 

4-     2.72     —  1.02009 

—  1 

- 

U0.090 

—  3.1*1 

—  117H03 

4-  3.20:1 

—171.79     -i:i.l:l 

4- ,'(931.21 

-f3oil.82     4-50.7820 

0 

+ 

3.31 

4-  0  4U 

4-     5  9 

—  0.4 

4-  12.24     4-  l-'*4 

4-  12.24 

4-     1.84     4-  0.98009 

+   1 

0.09 

—  0.01 

—    0.0!> 

4-  0.02 

—     0.51     —  0.08 

—     0.(10 

—     0.09     -j-   0.49512 

3,-  f. 

4- 

0.81 

—  0.04 

4-     27 

4-  0.2 

4-     3.0       —  0  1 

4-     S.5 

—    0.2      —  0.28814 

—  4 

+■ 

12i>9 

—  0.11  i 

+  ;i(;.9 

4-  3.8 

4.  34.5       —  3.(1 

4-  73.(1 

—     0.7       —  0.40478 

—  3 

+ 

2lHi.:)7 

—  0.04 

+(i01.l:t 

—  0.42 

4-717.94     4-  081 

-1-1535.81 

4-     0.115     —  0  (I8OOO 

—  2 

lij  3:t 

—  9.8,S 

-l.'i0.02 

4-lll.M 

—  190  :!3     —27.51 

—(1:15. 2:1 

—111.09     —  2  12559 

—  1 

+ 

■').ii.'i 

+   1.5ti 

+  l:i.S 

—  :).l 

4-   20.12  ;  4-  4.(15 

4-    1.0 

—     1.2     1  4-   1.88844 

0 

0.24 

—  0.07 

—    0.6S 

4-  0.15 

—     1  11      —  0.30 

—    1.11 

—     0.30     +  U.  05378 

4  -  « 

4- 

0.7-. 

—  0.07 

+     3.2 

4-  0.3 

-1-     3.4       —  0.2 

4-    0.7 

-     0.4    1  -  0.25249 

—  5 

4- 

9.-17 

—  0.79 

-f  38.:» 

4-  4.0 

4-  :is.5      —  3  9 

4-  70.4 

—   0.0   '  —  0:1:1777 

—  4 

4- 

Ill.l.i 

-  0.12 

4-441.7 

—  O.H 

4-5i2.2i    4-  o.:i5 

4-9(15.7 

—     (1.1     ;  —  O5I01I4 

—  3 

— 

;j8.fi8 

—  7.4t 

—  149  10 

4-22.:l7 

—  17-  15     —27  95 

—419.12 

—  74.3(1     —  1.(410; 

—  a 

4- 

."i.2a-. 

+   1.^9J 

4-  19.. 54 

—  5.50 

4-  25.:tO     4-  7.19 

—  603.3 

-185.4       4-25.:i910 

—  1 

0.3lJ 

—  0.17 

—     1.28 

4-  1.49 

—     1.90 

—  0.70 

—     1.21 

—     0.37     +  0.90210 

6,-7 

4- 

O.!!,"! 

-  0.07 

4-     3.5 

4-  0.4 

4-     3.5 

—  0.3 

4-     6.5 

—     0.5       —  0.22468 

—  0 

4- 

U.97 

—  0.59 

4-  3.5.1 

4-  3.4 

4-  3(1.7 

—  3.4 

4-  liO.9 

—     5.6       —  0.281184 

—  5 

4- 

(13.03 

—  O.K! 

4-315.3 

4-  0.1 

4-.3.54.S 

—  0.1 

4-012.0 

—     0.7       —  0  40804 

—  •'. 

— 

27.43 

—  5.32 

— I:tt4 

4-21.4 

—  1 5:1.1! 

1  -25.3 

—304.8 

_  64.  ti     ;  _  0.08U29 

—  3 

4- 

4.84 

4-  1.81 

4-  23.0 

—  7.0 

4-  27.82     4-  8.57 

4-  92.2 

■f  32.7    '  —  2  21043 

--  2 

— 

0.45 

—  0.24 

—     2.09 

4-  0.90 

—     2.77     —  1.14 

+     0.48 

4-     0.57 

4-  1.82073 

—  1 

■f 

0.03 

4-  0.02 

4-    0.12 

—  0.0(1 

4-    0.18    4-  0.09 

4-     0.15 

4-     0.07 

4-  0.0455 

«,-7 

4- 

4.9:1 

—  0.41 

-f  29  ,8 

4-2.7 

4-  31.5        —  2.9 

4-  -19.0 

—    4.4 

—  0.253S 

—  (1 

4- 

3li  lli 

-  0  17 

4-217.1 

4-  0.4 

4-2:111. IP       —  0.4 

4-387.4 

-     1.1 

—  0.3400 

—  5 

19.01 

—  3  74 

— Ul.ti 

4-18.8 

-125  7       -214 

-22:1.8 

—  40.0 

—  0.5152 

—  4 

4- 

4.11 

+    1..51I 

-f-  2:1.9 

—  7.8 

4-  27  7(1     4-  9.01! 

4-  (13  0 

4-  22.0 

—  1.0(12- 

—  3 

0.4i»(» 

-  0.271 

-     2  79 

4-  1  :io 

—     .3.42     —  1.55 

4-  4i!.40 

4-  20.01 

4-10  9273 

2 

+ 

0.03(i 

4-  0.02J 

4-     0.20 

—  0.13 

-f-     0.28     4-  0.10 

4-     0.14 

-i-     0.00 

4-  0.944:1 

7,—  8 

4- 

3.41 

-  0.28 

4-  24.0 

4-  2.1 

4-  25.5        —  2.5 

4-  37.8 

—     .3.5 

—  0  2267 

—  7 

4- 

2i>.!l(» 

—  0  M 

4-14H.4 

4-  0.4 

-f-lilii  0       —  0.(1 

4-245.3 

-     1.3 

—  0.21114 

—  « 

12.1III 

—  2.. 54 

—  811  2 

4-15.4 

—  98  11       —17  2 

—  1(13.1 

—  29.8 

—  0.411:1 

—  R 

4- 

3  40 

4-   1.31 

4-  2:!.o 

—  7.7 

+  2il.O     :  -f-  8.x 

4-  49.8 

4-    18.0 

—  0.(i9o8 

-  4 

0.4x7 

—  0.27ii 

-     3.25 

4-  i-':i 

—    :i.79    —  1.87 

—  12.83 

—     0.99 

—  2.311)8 

—  3 

4- 

0.044 

+  0.o:i4 

4-     0.28 

—  0.19 

4-    o.:t7    4-  0  24 

—     0.09 

—     0.12 

4-   '-7577 

8,-  9 

4- 

2.3.-. 

—  0.20 

4-  18.8 

4.  i.« 

4-  19  n    i  —  2.1 

4-  28.1 

—     1.2 

—  0  2032 

—  8 

4- 

12.20 

—  0.1:1 

4-  9ti  (1 

4-  o:i 

4-105.0     —  0.7 

4-1.54.8 

—     1.3 

—  o,2.'prio 

—  7 



8  M 

-  1.71 

—  (18. !t 

4-12.1 

-  75.5      -l:!.l 

—117.7 

—  21.3 

—  0.3423 

_  (i 

4- 

2>:9 

+■  1.05 

4-  21.0 

—  7.1 

4-  2:t.2      4-  8.1 

4-  40.0 

4-  14.7 

-  0  5205 

—  5 



0  45 

-0.2i 

—     3.5 

4-  1.8 

—     3.90  !  —  1  99 

—     8.8 

_     4.H       _  1. 0855    1 

—  4 

4- 

0.0411 

+  0.039 

4-   o.;t7 

—  0.2«1 

4-     0.45     4-  0.31 

—     4..50 

—     3.09 

-f- 12. 0958 

9.-  9 

4- 

7.0 

—  0.1 

4-  «3.4 

4-  0.1 

4-  (18.2       —  0.(1 

4-  9(1.8 

—     1.0 

—  0.2207 

—  8 

BO 

—  1.13 

—  51  9 

4.  9.1 

—  5(1  3     1—9.9 

—  84.0 

—  15.2 

—  0.29:11 

—  7 

4- 

2.117 

4-  o.si 

4-    18.1 

1(1.4 

4-19  8     1  4-   7.0 

4-  31.8 

4-   11.7 

—  0  4147 

—  6 

0  liO 

—  0.25 

-     3.4 

4-  1.9 

—     .1.78     —  2.02 

-     7.1 

—    4.1       —  0.7085    1 

—  5 

+ 

0.080 

4-  0.042 

4-    0.43 

—  0.32 

4-     0.51 

4-  0.3G 

4-     1.72 

4-     1.38 

—  2.4308 

10,-10 

4- 

4.1 

—  O.l 

4-  43. 

0 

4-  44.3 

—  0.8 

4-  fil. 

—    0.8 

—  0.2040 

—  9 

— 

3.9 

—  0.8 

-  40. 

4-  6. 

—  42.0 

—  7.0 

—  BO. 

—  11. 

—  0.2503 

—  8 

4- 

1.8 

+  O.li 

4-   14.7 

—  5.(1 

4-  1(1.1 

4-  (1.0 

4-  25.1 

4-     9.3 

—  0  3446 

—  7 

— 

0.3 

—  0.2 

—     3.1 

4-  1.9 

—     3.5 

—  1.9 

—     5.8 

—    3.4 

—  0  5259 

—  « 

4- 

0.0ft 

+  0.04 

4-     0.4(1 

—  0.37 

4-     0.53 

4-  0.40 

4-     1.1(1     4-     0.9« 

—  1.1092 

THE   on BIT  OF   URANUS 


fiS 


The  term  of  hmg  I\ri(Ml, 

From  the  cx[)rossioiis  for  tlio  perturbations  of  Uranus,  sijbsoqucntly  Riven,  it  will 
be  seen  that  several  of  the  terms  luive  very  large  coeffieieuls,  that  of  sin  (-/'— J/) 
being  nearly  an  entire  degree.  Tiie  magnitude  of  most  of  the  terms  in  which  I'  is 
even  arises  from  the  near  approach  to  commensurability  in  the  mean  motions  of 
the  two  planets.  Twice  the  mean  annual  motion  of  Neptune  exceeds  that  of 
Uranus  by  only  30:J".8.  The  elements  ef  the  orbits  of  botli  planets  will  there- 
fore, in  consequence  of  their  mutual  action,  be  affected  with  a  slow  oscillation, 
having  a  period  of  about  4'2(>6  years.  The  employment  of  these  large  terms  and 
the  great  inconveniences  to  which  they  will  give  rise,  especially  in  the  corrections 
of  the  elements  of  Uranus,  may  be  avoided  by  the  device  employed  in  the  theory 
of  Neptune,     The  following  arc  the  essential  featun^s  of  this  method: 

YJrat,  all  the  perHnbations  arising  from  that  portion  of  the  perturbntive  func- 
tion in  which  the  coefficient  of  the  time  is  2m'—  n  or  its  multiples  are  considered 
and  developed  as  perturbations  of  the  elements. 

Secondly,  tlic  arbitrary  constants  to  be  added  to  the  integrals  of  these  perturba- 
tions are  so  taken  that  the  perturbations  shall  vanish  at  the  epoch  1850.0. 

In  other  words,  the  perturbations  in  question  wiil  be  treated  as  producing  secular 
variations  of  the  elements  of  the  orbit,  only,  instead  of  being  developed  in  powers 
of  the  time,  these  variations  will  be  retained  in  their  rigorous  form. 

The  formuliE  for  the  conq)utations  of  the  perturbations  in  question,  arc  as 
follows : 


Let 


m 


m 


h  cos  («"  I' -\-il  -|-/(j'  +  j(j)  =      cos  iV 


be  any  term  of  the  pcrturbativc  function,  7i  being  a  function  of  a,  c,  and  a. 

sin  4,  =  e 
g  =  cos  4.  tan  J  4' 


For  each  such  term,  compute 


v  = 

n 

i'u'-\-  ill 

A  = 

2  !h 

L=^ 

cW  = 

ens  4/  ^ 

E  = 

—  ''('.'/+ J  ">t  4) 

T  = 

1  = 

l,'^  +  (/+y)a/«. 

M 


THE  OliniT  OP   U  RAN  US. 


The  corresponding  perturbations  of  tlic  demonts  mny  then  bo  put  into  the  form 

6  log  a  =  MvA  cos  N-\-  5»o» 

il  =  Mi'L  »ii\  N -\- i(o, 
ehn        =  Mve  IVsin  JV-f  efi7i„ 

he         =  MvE  cos  N  -\-  Scq, 

iy  =  Mi'f  COS  N-{-  fyo, 
tan  y  5t  =  Mv  T  sin  iV  -\-  tun  y^To. 

Here,  5»)o,  3?o»  etc.,  arc  arbitrary  constants  so  taken  that  5  log  a,  3?,  etc.,  shall 
Tanish  at  the  fundamental  epoch. 

All  the  terms  depending  on  the  same  values  of  i'  and  /  are  to  be  combined  into 
a  single  one.  And  it  will  save  labor  to  make  this  combination  at  as  early  u 
stage  as  possible  in  the  computation;  that  is,  to  nuiltiply  the  various  values  of  /j, 

S^Jl    ^JL,  _'    ,  and  27 by  the  sines  and  cosines  of  I'u' -t-iw,  and  afterward  proceed 
Sv     6e      Sa 

with  the  sums  of  the  products  according  to  the  proper  modification  of  the  formulir. 

Thus  arc  obtained  the  following  long  period  perturbations  of  the  elements  of 

Uranus : 


—  3474.3-2  sin  (3/' 
+    im.T*  mi  {At— '2i/)  — 

—  8.97  sin  (()/'—  Sij)  + 
4-  0.()4sin(W— 4J/)  — 
4-consfant=3320".lS. 


r/)-j- 180.10  cos  (2/'—    fj) 

54.10  cos  (4/'— 2//) 

5.03  cos  (()/'—  iiij) 

0.53  cos  (8/'—  4fj) 


eSn  =  —  484.9()  sin  (2/'—  //)  + 
+  38.00  sin  (4/'— 2J/)  — 
—  3.01  sin  (r)/'—3</) -f 
4-  0.33  sin  (8/'— 4.7)  — 
-f  constant  =  405".  23. 


0.73  cos  (2/' —  g) 
7.0()  cos  (4/'—  2y) 
1.38  cos  (()/'— 3.7) 
0. 15  cos  (S/'— 4//) 


=  _    484.21  cos  (2?'—   «/)  — 
+      38.21  cos  (4/'—  2j/)  4- 

—  3.01  cos  (0/'—3f/) — 

4-        0.33cos(8/'— 4r/)4- 

—  constant  =  158".  59. 

3i)=4-       2277  cos  (2r—    .7)  + 

—  198  cos  (4/'—  2)/)  — 
4-  18cos(G/'— 3(/)  + 
4"  constant  =:  630. 


0.29  sin  (2?'—  .7) 
7.10  sin  (4/'— 2j/) 
i.40sin(fi/'— 3.7) 
(I  15  sin  (8/'— 4(/) 

120  sin(2r—    g) 

78  sin  (4/'—  2.7) 

7  sin  (0/' —  3.y) 


The  variations  of  the  elements  wliich  fix  the  position  of  the  jdane  of  the  orbit 
arc  here  omitted,  because  their  nature  is  such  that  it  is  indifferent  in  which  form 
they  are  developed. 

These  expressions  are  reduced  to  perturbations  of  the  co-ordinates  by  the  follow- 


•- 


' 


THE   OUDIT   OP   UTIANUS. 


67 


ing  forniultr.     Express  the  usunl  dcvcloitmcnts  of  tlic  longitude  nnd  logarithm  of 

radius  vector  in  the  form 

«  =  ?  -|-  2  V,  sin  i(j  I 

p  =  »  -|-  2  ii,  cos  iff. 

Put  also 

e  .' 

;„  OR, 

oe 


ir,  =  -'  R, 


Express  any  set  of  corresponding  terms  of  the  preceding  perturbations  in  the  form 


}l  z=  L,  sin  N-\-L,  con  N 
e}.l  —  chn  =  F.  sin  N-\-F,  cos  N. 
le  =  Eo  cos  N-\-  E,  sin  N\ 
hv  =  A,  cos  iV+  -4,  sin  N. 
Wc  shall  then  have 

25r  =  S  ( r\  F.  +  V\  E,.)  sin  (A'  +  »;/)  +  2  ( V",  F.  -  V,  E,)  sin  (iS^-  ij) 
+  V  (  V\  F,  -  V\  E.)  cos  (iV+  iy)  +  li  (  V'\  /'„  +  1"«  ^'.)  cos  (A'-  ig) 
+  2ii 

2:V  =  2  (/?,  E„  -  /?",  /^,)  cos  (A^+  ig)  -\-  2  {K,  E„  +  iT,  /-.)  cos(A^-  vj) 
+  V  (/^,  E.  +  7?",  /',)  sin  (Ar+  tV)  +  2  (/f'<  E.  -  R\  /',)  sin(A'-  ij) 

-\-  2)v 

The  numerical  values  of  T",  F,  /?',  and  liT  arc  as  follows : 


r,  =   1.098:35 

F3=  o.ini;} 

F,=  0.00714 
F4  =  0.00044 
i?'.,=  +  0.0'i;)48 
jT,=  — 0.90703 
i?*,  =  —  0.070'20 
71"!.=  — 0.004GG 


F',  = 

F'3  = 

F'4  = 


1.99945 
0.11722 
0  00714 
0.00044 


i?',  =  +  0.99917 
Jl\  =  +  0.07030 
i^a  =  -j-  0.00467 


The  final  results  of  the  entire  computations  arc  given  in  the  following  table: 
In  the  columns  ii;„  wc  have  the  complete  perturbations  of  the  longitude  computed 
by  the  direct  method  from  the  values  of  M,  ^j^ ,  (?,  etc.,  already  given.      Next 

wc  have,  under  the  caption  t'r,,  the  perturbations  of  the  true  longitude  deduced 
from  the  long  period  perturbations  of  the  elements,  as  set  forth  in  the  last  para- 
graph, omitting  the  constants  added  to  the  perturbations.     Under  hv^  we  have  the 

8        April,  1873. 


08 


THE   ORBIT  OF   URAXUS. 


perturbations  of  the  longitude  deduced  from  all  tlie  remaining  terms  of  the  per- 
turbutiuns  of  the  elements.  The  sum  of  the  columns  ii'j  and  hvj  shows  the  entire 
perturbutionr,  computed  by  the  method  of  variation  of  elements.  Thus,  in  })Vi 
and  hv-i  -f-  ^^3  we  have  two  complete  sets  of  perturbations  computed  by  methods 
entirely  independent.  The  diftcrences  of  the  results,  expressed  in  thousandths  of 
a  second,  arc  given  in  the  last  two  columns  of  the  table. 

Tliis  comparison  gives  rise  to  remarks  similar  to  those  suggested  by  the  per- 
turbations of  Saturn  computed  by  the  same  methods.  The  only  terms  in  wliicli 
the  difference  of  results  amounts  to  w  much  as  one-tenth  of  a  second  are  those  of 
very  long  period,  and  those  very  nearly  the  period  of  Ij  anus,  wliere  a  mon;  accu- 
rate vaiiK>  is  not  at  present  of  great  importance,  because  the  error  will  be  com- 
pensated by  the  corrections  of  the  element  during  several  centuries. 


Pertuhbatioss  or  tii« 

LoNOITUDE  OF  Uh.\NUH  PIIODUCED  BY  NeI'TUNB. 

poiH-y. 

< 

"i 

ai'. 

«i 

J 

Discri 

f  'J 

sin 

cos 

sin 

cua 

sill 

cos 

sin 

COS 

0,     0 

— 0.4  202  < 

—1 

» 

on.')  lit 

—  1 

.isl  m 

—a 

— 

004  (1/ 

—0.009  nt 

0,     0 

_\ 

+ 

O.fittT 

— 

o.oss 

—•2 

+ 

o.o4(; 

— 

0.005 

—i 

+ 

o.ooa 

1,— T 

-f 

0.147 



0.001 

.... 

+  0.140 

—0.002 

1 

1 

2 

+ 

2.r)0i» 

— 

0.01 !) 

.... 

4-  2.509 

—0.010 

0 

9 

—  1 

+ 

3!t.(i5S 

— 

o.oso 

.... 

.... 

+39.073 

—0.0.^1 

15 

1 

0 

+ 

4. 2;-)  7 

— 

o.r>ii 



+   4  249 

— 0.47S 

S 

33 

—1 

+ 

0.2S0 

— 

o.o;i2 

...  * 

+  0.275 

—0.032 

6 

0 

-2 

+ 

0.017 

— 

«.002 

2,-4 

+ 

2.ft7S 

+ 

0.0  Jf 

+      2.  so 

-f     0.03 

+  o.oos 

—0.002 

10 

1 

:I 

+ 

41I.01.5 

+ 

o.4l:i 

+     47.2:' 

+     0.4  4 

+   1.797 

—0.021 

2 

0 

—2 

+ 

H40.!t;{ 

+ 

7.:1HS 

-f   HO5.04 

+     7.43 

+  35. .355 

— .1.028 

0.5 

14 

—I 

—347-).  4 

+ 

ISO., 'to 

-3474. 32      +1S0.IO 

-  -  0.700 

+  0.095 

3S0 

ir>5 

0 

— 

i(;2.o7 

+ 

S.OI 

—  101.90 

-f     S.OI 

—  0.007 

+0.015 

43 

15 

—1 

— 

9.447 

+ 

0.408 

—       9,50 

+     0.47 

3,— :> 

__ 

0,070 

0.000 

... 

—  0.077 

-0.003 

1 

.T 

—4 



l.ir.j 

0.000 

■  . , , 

—  1.1,53 

— O.dll 

9 

11 

—a 



17.2S(! 

+ 

0.0-2S 

.... 

— 17  2H5 

+  0.2-29 

1 

1 

—2 

— 

22.0S5 

+ 

4.o:i7 

.... 

-2-2.077 

+  4.(t-J0 

H 

17 

—1 



o,(i7:{ 

+ 

0.0S2 

•   .   .   . 

.  t  • . 

—  O.0S2 

+0.079 

9 

3 

0 

— 

0.0U7 

+ 

0.000 

4.-fi 

. 

0.0:50 

+ 

0.00-2 

—     0.01.) 

-f   O.OOv 

—  0.027 

—0.003 

0 

.1 

—5 



o.ftr.s 

+ 

o.o:i7 

—     0.25 

+  0.04 

—  0.315 

— 0.007 

7 

4 

—4 

— . 

7.!tr,8 

+ 

0.7')0 

—     4. OS 

4-   0.08 

—  3. 90S 

+  0.059 

20 

11 

—3 

..— 

7.'>.oo 

+ 

12.s:i2 

—  09.55 

+  11.07 

—  5.7.13 

-+I<I07 

2S3 

95 

—2 

+ 

14fi.7S 

5I.21S 

f  140.72 

—54.10 

+  0.1-20 

—0.079 

00 

39 

—I 

+ 

6.900 

— 

2.579 

+     O.Sl 

—  2.03 

u 


THE   ORBIT   OP   URANUS. 


69 


I'EKTUKUATIONS  of  TilK 

LuNuiTUDe — Voiilinueu 

• 

««, 

««, 

tv 

1 

tiiacrc 

>ancj. 

i'    9 

sin 
// 

COS 

Biti 

cos 

siu 

COS 

sin 

COB 

b.~  7 

—0.000 

0.000 

.... 

—0.015 

—0.002 

C 

2 

—  (! 

—0.103 

+  0.001! 

,  .  ,  , 

—0.113 

—0.004 

10 

10 

—  5 

— 0.!»«(! 

—0.043 

.... 

—0.004 

—0.042 

K 

0 

—  4 

-f3.;jfifi 

—0.(170 

t  .  •  • 

+3.370 

— o.(;(i2 

4 

8 

—  3 

+3.210 

— l.lfiO 

•  .  .  • 

+  3.227 

—  l.lHfi 

17 

17 

—  2 

+0.077 

—0  017 

.... 

.... 

+  0.075 

—0.002 

2 

15 

—  1 

-fO.OOD 

—0.001 

«.—  7 

—0  050 

-0.004 

0.00 

0.00 

—0,054 

—0.002 

4 

2 

—  fi 

— 0.3«7 

—0.020 

+0.02 

—0.01 

—0.423 

—(,.013 

in 

3 

—  5 

-fl.2fil 

—0.320 

+  0.40 

—0.14 

+0.S.55 

— o.i(;o 

« 

11 

—  i 

+  7.7HI 

— 2.S25 

+o.«o 

—2.54 

+0.S57 

— 0.3IH 

124 

33 

—  :i 

—0.025 

+5.073 

— S.07 

+5.03 

+0.017 

—0.022 

72 

C6 

0 

—0.437 

40.240 

—0.42 

+  o.2i; 

T,—  « 

—0.027 

—0.002 

—  7 

— o.isn 

—0.003 

.... 

—0.180 

—0.009 

3 

fi 

—  fi 

+0.272 

— o.om 

+0.2(iO 

—0.047 

12 

1 

_  f> 

—0.571 

-<  0.217 

— ().53S 

+0.202 

33 

15 

—  4 

—0.450 

+  0.250 

—0.410 

f 0.255 

40 

5 

—  a 

—0.010 

-I  0.005 

«,—  i» 

-(t.(ll3 

0.000 

—  H 

— 0.0!(4 

—0.002 

— 0.003 

—  <>  007 

0 

5 

—  7 

.)  0  125 

—0,022 

-0.115 

-       •24 

10 

2 

—  fi 

—0.104 

■\  0,OM2 

— 0.03S 

4  0.017 

—0. 1 45 

+  0.055 

11 

10 

—  5 

— O.H30 

1  (»,4t'.3 

— O.fifi 

-t-0.30 

—0.110 

+  0.007 

(19 

06 

—  4 

+  O.G47 

—0,501 

-fO.«4 

—0.53 

9,—  It 

—0,040 

—0.001 

— 0.04^^ 

—0.005 

8 

4 

—  H 

+  0.0fi3 

— O.Oll 

•   •   •   ■ 

+  0,002 

—0,011 

fil 

0 

—  7 

— 0.055 

+  0.020 

.  .  .  • 

•   .   •   • 

—0,050 

-t  0.025 

4 

5 

—  0 

+  0.OHO 

— o.o4;t 

•   .   •   • 

+0.0«3 

—0,054 

3 

5 

_^        r 

-f  0.050 

—0,050 

10—10 

—0.020 

0.0(»0 

.... 

—0.025 

—0.005 

5 

& 

—  0 

4  0.035 

—0. 005 

—  8 

—0.027 

— OOIO 

—  7 

f0.02ti 

—0,0  IT 

—  fi 

+  0  002 

—0,070 

Thoporturhatioiisof  thol()p:!>''ith"isof  the  radin.s  vrctor  nro  givon  in  nform  similar 
to  thoso  of  the  loDf^itiulc.  lliwlcr  ^p,  we  hiive  Mio  coinpUat'  perturbution.  Under 
^p.j  the  I'ffoct  of  tlic  perturbations  of  long  period.  But  luuler  ipa  wc  have  only 
the  difference  between  ^p,  and  ^p„  it  being  deemed  nnnece-ssary  to  present  in  full 
the  perturbations  of  the  radius  vector  as  computed  by  the  other  method.  Ap,  beinir 
employed  in  computing  ^»,  may,  in  fact,  be  regarded  as  completely  checked  ty  its 
affording  a  correct  value  of  the  latter. 

In  the  last  two  columns  f^p,.  is  reduced  to  common  logarithsns  by  multiplying  the 
coefKcieuts  by  the  modulus  0.  i;}4'.'!)4. 


60 


THE   OUHIT   OF   URANUS. 


I'KKTL'KUATIU.NS   Ut  TIIK    LodAlllTIIM    of   TIIK    KAIHI.S   VKCTKIl   OK    UlU.Nl  8    IMUH-I CKU 

IIY    KkI'TINK. 


f      0 


0,     0 


1,-3 
2 

—  1 

0 

+  » 

2,-4 

— ;i 
2 

—I 

0 

3,— ."S 
—4 

_;{ 
-J 

—1 

4.-fi 

—4 


_4 

_3 

— r. 
—r. 

—4 

_3 

7,-7 
— f. 
— 5 
—4 

8,— s 
—7 
—0 
—5 

9,-9 
_s 
—7 
— ••> 


»P. 


-f  i:i8 


+  4 
-f  fi8 
+.')'2:i 

—  (IS 

—  8 

+      «4 

+i4i(; 
+-.»(i02r) 

-I  IC.fiS 
4-3!)  1 -J 

—  3 

—  3S 
—■.•->  7 
— 1>S4 

-I  ir> 
_    1 

—  10 
—•.'.".4 

— nr.tt 

—  143 

—  1(17 

—  4 

—  40 
-1-103 

-f  3'J 


—  17 

+  ■«•-' 

-f  ISO 

+  i:5 

—  s 
4  11 

—  17 


-  4 

-t-  ;i 

—  7 

—  19 

2 

4  ■■! 
2 

+  a 


0 
0 

+  > 

—  c. 

—  1 

—  1 

—  \i 

—  179 
-I  IK! 

+  194 

0 
0 

—  (J 

—at 
4    2 

0 

—  1 

oo 

— :ioo 

—  01 

—  (■.-' 

0 

+  > 

+  -.MI 
+  1& 

0 

4-  1 
4I«» 

+.ir. 

+  « 

0 

4  i 

—  7 

—  3 

0 

4-  1 

—  3 

—  II 

0 
0 

—  1 

+  a 


*p« 


-f  92 
+  1374 
-1-19190 
-f  1720 
-f  39J7 


1 

tf. 

Hill 

cort 

—     1 

+      -' 

—  12 

4  •«•-' 

—  ISO 

4-r.35 

4  120 

—  03 

4  194 

—  ir. 

s 

—  lis 

—  KiSi 

—  ir.4 


4-  10 

4-1C.4 

4-  H 


—  20 
'.>><2 

—  70 

—  (U 


4 

01 


—  11 

—  13(i 

—  7s 
4-  II 


17 

32 

I(i 

1 


0 
0 

+    « 

—  4 

0 


Mif 


—  I 
2 

—  IS 

4   '•• 
+  a 


0 

+  I 
4-  6 

4-  * 
-\-  3 


1 
COS       1 

4- 

00 

4- 

2 

4 

30 

I-- 

27 

30 

— 

3 

+ 

I 

4- 

IS 

L.  •»■!.>    1 



27 



7 



1 
17 

— ; 

!29 

— 

24 

+ 

7 

r. 

_ 

ri9 



34 

4- 

1 



17 

•f 

45 

4- 

17 

—  1 

—  7 
+  U 
4  7 

0 

—  3 

(■  ■'' 

—  7 


+     I 
—     1 

4-    I 


2  ■ 
3 

«  i 
I 
1 


0 
0 
0 

—  3 
U 

0 
0 
0 

2 

U 

0 
0 

—  3 
—23 
4    1 


0 

—  1 

—  H 
4-   4 

4-  I 

0 
0 

4-  « 
-f  (i 

0 
0 

4  :i 
4  a 
4-  I 

0 


0 

(I 
_  I 

—  5 

0 
0 
0 

4-  » 


I   ( 


u. 


•  -T^'iHifiiiflfrarltiiMf'W 


1 


i 


1 1, 


T 


A 


u 


■) 


THE   OKBIT   OF    UUANUS. 


61 


reilTI.KIIATIUNH  ur  TIIK  liATlTl'DK  PUODL'CEI)  BV  NciTUNB.                                                      1 

dit       >»' 

J  '*    (liD  .V 

2f 

I  ('■+»  r 

'/"J 

vr       "i 

XA  "iii.V 

J=«'.„X 

ik 

«T 

tfl 

1'      1 

COS 

liu 

■Id 

OM 

Hill 

ooa 

ilu 

001 

•in 

CM 

CO. 

■In 
// 

sin 

r, 

0     0 

— 21.G5 

0 

0 

0 

0 

0.00 

0 

0 

0     -.012 

1 

+  l.ll^,^-2.7:l'—  4.0(1— l.tiUf  .01— (Mil 

—  3.!>ll— l.r.7i—  0.008 

+0.021 

+0.030 

—0.012 

+  .080 +.020 

2 

+  li».32— U.3!)+1<j.32,  +  1).:J1»—  .01      0 

II                                                       1                         1 

+  111.31  +11.31) 

—  0.073 

— 0.03ti 

—0.073 

+0.(lJ)i 

+  .01(i— .002 

1,-3 

+  14..'.^+7.0  Ll4.7  1+7.11  1     0      !     0 

-14  7    +7.0 

+0.045 

—0.022 

—0  045 

-0.022 

+.0(17  —.(102 

'J 

—  1.1«  --J.liO -f  4.li4—'J.40— 0.07+0.02 

+  3 1)7      2.4 

— 0.(  (1(1 

+  0.014 

+0  (120 

+  0.012 

+.201  —.04(1 

—1 

—34  2.1      0      1       0      1     0       +1.7tJ      0 

+  1.7(i      0  .. 

-0.540 

0 

+  0  (128 

0 

—  (108  —  (i:(7 

0 

+  :i.74  +'J.:i2—  2.H0  --  1.48  -0.24      0 

—  3.(14-1.48 

-  (1.057 

+0.o3.'i 

+0  04il 

—0.022 

+  .3(i4+.(if4 

+» 

+  1S.77— 7.h7+ll).20+7.7i     0          0 

+l(i.2i  +7.75 

1 

—0.081 

— (1.040 

—0,083 

+0.040 

+.OU(i  +.007 

1 

2,-4 

+  10.52+.1.0(!— lO.K  '+6.0  !     0           0 

-lO.tf    +5.0 

+0.027 

—0.013 

—0  027 

—0013 

+  .002— .001 

—3 

—  2.tiO— 2.38+  3.(ji— 2.28 +ii.3ii+ 0.04 

+  4.01  -2.24 

—  0.010 

+  0.(1(1!) 

+  (1  Olil 

+  0.(8'!) 

+  .0(j2-.(il4 

—a 

— 3:l.;j»'     0      1       0      >     0       -f  "J.^.s —ii.dl 

-I-  D.i-M  — (i.dl    _  (i.-i;4         0 

+  0  ('78 

0 

+  .008  —  (104 

—1 

+  &.M'+2.04—  2.12—1  04  — l..');i+0.04 

—  3.115    -  1.(10  t— 2.283)  [  +  0.K12]  t  +  1.431)  [-0.31)3] 

+  .32(1 +.075 

u 

+  18.7U  — a..V2  +l«.5li  +i>.;i2+0.08      0 

+  ll).(i4+l).32 

1 

-0.143 

—0.072 

— 0.04U      +0.071 

1 

+  .002       0 

3,-5 

+  7.4  '+3.ii  !  -  7.4    +3..'.  ;     0          0 

-  7.4  +;i.5 

+0.017 

—  O.OOH 

1 
—  0(117       —0(108 

+  002       0 

—4 

—  3.011  _2.0ii+  3.12      2.0  +0.48+0.05 

+  3.(!()  -2.0 

—  0.010 

4-0. oou 

4  0(111 

+  OI8l(j 

+  .oi)_.lli4 

—3 

—  21.4  j     0      1       0       10       +7.87      0 

+  7.87      0 

-0.128 

0 

40  041        0 

-.(K^+.(l4l 

•1 

—  •* 

+  ii.5h+|.7.l—  1.20— 0..14— 1.08  +o.-J(!-_  3.18—0.28 

+0.108 

—  0  0211 

-  0  (153 

+  0(104 

+  303 +.074 

—  1 

+  l:l.ii(;  —7.28  +14.tih  +7.(KI  +0.1^  — 0.04  —14  8il  +l!.!lii 

—0.1  111) 

—  0.1(1(1 

—0  217 

+  0.101 

—  .{190  — .icu 

0 

—  1.61  +0.22  —  l.Sl  —0.22      0           0 

1             1               i             1 

—  l.il  -0.22 

i 

+  0.008 

+0.001 

+  o(Ki» 

—  (1.001 

—  .013       u 

1 

4,-5 

-  2I». -1.(13+  2.53  ^1.7    fO..'.        0 

+  3.0    -1.7 

— (I.1K>8 

+0.(814 

+  0  (08 

+  0  0(14 

4-.  022 —.004 

—4 

—  17.4        0      1       0           0       +''-8   '     0 

+  5.8        0 

—  0.070 

0 

+  0  023 

0 

—Mill-  +.(Ki(i 

—3 

+  (1.;U  +  I.34—  O..'!!)  —0.22      1.07  +0.:10  —  2. .'XI  +0.08    -(0.(151 

— O.OII 

(1  (I'ji     —0  001 

4-.(l4^ -i-.olO 

-i 

+  11.42-  .'..37+10  57  f  5.04 +0.25      0.07  f  10.82 +4.117  I     1.8Vj)  (     l  (i5i;|  l~2  1J71  l+o  1(77) 

—  (147       0 

—  I 

—  1.72+0.32—  1.72    -0.32—0.112-1  '(.ua  _  1.74— o.iHi 

1            1              1            1            1  '          1              1 

+  0.013       +0.(KC 

+0.013    1    — 0.C02 

—  001       0 

8,-e 

-  2.4    — 1.4o'+   1.8    -  1.4   +0.B       0       +  2.3    -  1.4 

—0.00.11 

+0.0(13 

+  0(105   '  +0(103 

+  O12-.003 

—5 

—  12.(1        (1      1       U           (1       -f-4.0        0      1+  4.0        0 

— (i.(i:m 

(I 

+  0  013   j        0 

—.012       0 

-4 

+  5.5h +1.(11-  0.22 -11.04   -  1.75+11.28-  1.117+0.24 

+  (i.o3(i 

-  0.(8(5 

-   (1  olll       —0  (Nil 

—  1(0  — (128 

-3 

+  (1.25— 3. hU+  7.4.1  f3  5l+0.30— (i.('ll+   7.73+3.45 

+('.1('7 

+  0.0(17 

+0  l:ii 

—0  o51( 

—  (i;*4.  i,(il 

-2 

—  l.li(i+0.33—  1. (id —0.33— 0.(13 +0.01  —  i.iiU— 0.32 

III                           i 

+0.024 

+0.wi 

+0.(iJ6 

— 0  (104 

+  018        (1 

(J,-6 

—  8.4  :     0             n      '     n       +2.8        0       +  2.80      0 

-  0.022 

0 

+0(Ml7 

0 

-did  .(-.(812 

— S 

+  4.(13  +0.7(1  —  0.03  4-0.04  -  1.4(1  +0.24—  1.40  +0.28 

+0  011) 

—(1.(8)3 

—  0  (Kill 

-  0(01 

—  (128  — .(H18 

—» 

+  4.(13  -   •,;.71(  +  5.15  +2.45  +0.31  —0.  lo  +   5.4il  +2  35 

+o.o3t 

+  0.(122      +0  045 

—0  (111) 

—  Old       0 

—3 

—  I.4J  +0.31  —  1.45  —0.31  —0.04  -(-0.02  —  1.41)  -0.21i  l+O.llHi)  [  +  0.(141]  l  +  "  IW]  I— 0.038 J 

+  .015^      0 

7,-7 

—  S.(!        0 

1 

0     1    0     +1.1)      0     '+  i.n      0 

-0.012 

0 

+0.(^4 

0 

_.(Kl6+  dOl 

— tf   4-  .•;.ii:l -f  (i.iii 

0          0      —1.17  +0.2(1—  1.17+o.-'(i 

40.('12 

-    (1.002 

—  O  001 

—0  (ml 

_  (108  — ,(KI4 

-8 

+  2.411—2.(10+  3.52 +  1.08 +(1.30 —(i.lo-(-  3.82+1. 5« 

+  0.013 

+  (i.on 

4'i.i2ii 

— 0  (Hih 

+  .014      0 

-4 

-  1.111 +0.27—  1.1'J -0.27 -0.04 +0.02—  1.23.-0.25 

-0.021 

-(1.(05 

—0.022 

+  0.004 

+  .(8)2,      0 

1 

8,-8 

—  4.0  !     0             0           0       +1.3  '     0      !f   13        0 

-0.(818 

0 

+0.003 

0 

—  (Ki3  +.002 

—7 

+  2.711+0  50       0           0       —d.ltl  +0.1(1 

-  O.lil  +0.1(1 

+  0.007 

—0.002 

+  (1.1K12 

0 

_,(Ki.i— .ooa 

— « 

+  1.411-1.41+  2.3k +1.14 +0.27 -o.O!" 

1-  2.(15  +1.05 

+  (l.l«lli 

+  (l.(Nl>i 

+  0.011 

—  0(104 

+  (814       0 

—5 

—  0.1(7  +0.22—  (i.a7  — 0.22      0       +0.02—  0.U7— 0.20 

1               ,             .             1             1 

-0.(818 

-0.(8)2 

—0.(8(8 

+0  (Ki2 

+  (8)2       0 

1 

b,fular  Utriii,     M  =4- I "25  7' 

H  =  -  1  ■  -i:,  r  .-.HI  T 

Tlio  terms  iV.-  niul  fir;,  wlii(  h  iirc  inclosed  in  linieKets,  are  of  very  Ion;;  period, 
niul  arc  tliiTeCore  omitted  in  I'urminy  tiie  valius  of  yj  in  tiie  lii>t  two  (ohimns 


«9 


THE   ORBIT  OP   URANUS. 


Ihrlu)/ia!ioit8  praluccd  by  Jupiter. 

The  scries  in  which  these  pertmhations  are  expressed  converge  so  rapidly  that 
I  deem  it  unnecessary  to  present  tiie  details  of  the  computation.  Thoy  have  been 
computed  by  both  methods,  and  the  separate  and  independent  results  are  given  in 
the  following  table,  where  ht\  rcj..  scnts  the  perturbations  computed  by  the  method 
developed  in  Chapter  I,  and  it „  those  computed  by  the  method  of  varia.jn  of 
elements. 

The  apparently  large  discrepancy  between  the  coefficients  multiplied  by  the  time 
arises  from  the  circumstances  that  in  the  form  of  development  the  mean  motion, 
and  hence  the  mean  anomalj  ^j)pears  affected  by  the  perturbation  fiV'.Ql.  Accord- 
ingly when  we  enter  the  table  which  gives  the  true  longitude  in  terms  of  the  mean 
anonudy  in  the  form 

r  =  ?  4-  2c  sin  (/  —  «)-{-  etc., 

we  may  consider  this  quantity  31".2<  as  a  secular  variation  of  /  —  n  produc'ar  in  v 
the  term 

it)=^G2'.4c^cos(/  — 7t). 

In  rr,  this  term  is  left  in  its  primitive  form,  while  in  h^^  t''f"  value  of/  is  supposed 
to  include  this  term,  and  the  secular  terms  are  only  those  which  arise  from  the 
secular  variation  of  the  eccentricity  and  perihelion. 

It  is  also  to  be  remark«'d  tliat  the  terms  which  are  inde])endent  of  the  mean 
longitude  of  Jupiter,  or  thos(>  in  wliich  /' =  0,  are  not  comparable,  as  they  corre- 
spond to  slightly  diilurent  <  lliptic  elements  ia  tltu  tuu  theories. 


th 


THE   ORBIT   OF  URANUS. 


63 


I'EUTIRUATIONS  or  UllASLS  BY 

JlI'ITKR.                                                             1 

»y, 

t 

f. 

Uiff. 

C08  ^tf 

3l»p          1 

9    f 

»iii 

COS 

sin 

COS 

sin 

1 
COS 

coa 

sia 

cuu 

sin 

0,     0 

1 
2 

a 

—  0.  IC.Od/ 

—  0.010/1/ 

+ai.2ii(;< 

+a7.5S5n/ 
+  2.207/1/ 
+  0.135/1/ 

—  0.H;22/t/ 

—  0.00!t5/i/ 

—  O.OOOti/i/ 

+31.1ftH2/ 

—  1.540(;/i/ 

—  O.OSOO/i/ 

—  0.0054/1/ 

a 

0 
0 

0,     0 
2 

a 

+  25.«57 
+   I.a!l7 
+  0.073 

—  i.8.59    —  l.aVr, 
_  o.o«7    —  o.oao 

—  0.005             

+  i.3(;i 

+  0.080 

... 

— lOOS!) 
—401.5 

—   10 

—4387. 
—  213.7 

—  14.2 

—  0.8 

0 

—  0.8 

+  0.5 

0 

j 

—  32.8+   1.1 

—  1.8|       0 

—1,-1 

0 

1 

2 

a 

+  0.027 
+   1.2(!9 

— 5a.oc.' 

—  a.4;t'i 

—  0.14^ 

+  0.00!)     +  0.017 
+  0.002     +   1.232 
.  O.OOJ     — 53.0S4 
.-  0.0112     —  a.5tl5 
—  0.047     —  0.104 

1 

+  0.011 
+    0.001 

—  0.002 

—  O.OHJ 

—  0.050 

10 
37 
20 
70 

in 

2 

1 

2 

10 

a 

0.0—  0.5 

_  5«.H_  0.5 

+  25H5.7  +  0.1 

+  l!)5.a+  5.0 

+   n.\+  2.3 

0.0 

—  25.0 

+  1127.2 

+    82.  s 

+     7.4 

—  0.2 

—  0.2 

+  2.2 
+   1.0 

0.-2 

1 

SI 

a 

4 

—  0.027 
+   1.1S2 
+  0.277 
+   0.074 
+   0.015 

—  0.011     —0.031 
+  0.515     +   1.170 

+  O.oao    +  o.2r,a 

—  0.005      +   0.0X3 

—  0.003      +  0.014 

—  0.014 
+   0.515 
+   0.037 

—  O.OOS 

—  O.ooa 

4 

I'i 

a 

0 

1 

3 
0 

+      1.4—  0.0 
—  50.5  +24. !• 
+     8.5  +    l.H 
-j-     4.2  +   0.5 
+     0.0  +   0.2 

+  O.f. 
-24.0 
+  3.7 
+   1.8 
+  0.4 

—  0.3 
+  10.7 

+  l.C, 
+  0.8 
+   0.2 

1,— a 

2 

.1 

4 
5 

—  0  ()a2 

—  0.005 
+   0.025 
+  0.01 1 

+  o.ooa 

_  0.034     —  0.025 
0         _  0.005 
0         +   0.015 

—  0.001      +   0.037 

0       —  0.017 

—  0.025 

—  0.001 

0 

—  0.010 
+  0.004 

7 

0 

10 

2ti 

20 

!) 
1 
0 

n 

* 

I'KIITIRII.^TKINS  OP  TIIK  LaTITI  l)K.                                                                         1 

U      I' 

ih 

»1 

tfl 

t    l' 

0    0 
2 

a 

—2—1 
—  1 
0 

1 
2 

— ;v-2 

0 
2 

Hill 
'/ 

1  0.071 
—0  012 
—O.ooa 

—O.OOfi 
— 0.2!t7 
—0.1  CI 
\2.i\\\ 
4  0.070 

4  0.055 
+  0.002 

—0  no 

— 0.014 

COS 

'/ 

—0.011 
—0.075 
— (».0I2 

40,012 
4  l.!»l!) 
— 0. 101 
0 

+  tMI21 

—0.088 
— It.Oll 
-U.O.'.O 
0 

COS 

// 

40.030 

—  0.012 
— (».003 

40.000 

4(».2!i7 

— O.IS'.t 

4  2. "128 
-j  o.oco 

—0.055 
—0.002 
—0.10!) 
—0.011 

Kill 
'/ 

4  o.oac. 

)  0.075 

40.012 

4  0.012 
-1  1  '.no 

— 0.1  !I0 
0 

—0.002 

_0.0><8 
—0.011 
1  O.OJfi 
0 

C(I8 
// 

-1  .oi;o 

—  .010 
—.002 

-(  .005 

4.001 

— .4'.14 
— .021 
+  .002 

4  .010 
■\  .017 

— .00!t 
—.002 

sill 

// 

-4  .or.5 
4.047 
—.007 

0 

—.004 
—.003 
4  .420 
—  .013 

4.019 

—.003 

—.004 

(  .002 

0 

Sr.uhir  l.riiis   "  *'■            \\\Z 

««,)       —  1.14  1  cos  V 

64 


TUE   OKUIT   OF   UUANUS. 


CIIAPTKll   IV. 
TERMS  OF  THE  SECOND  OltDKll  I'UOUUCKl)  IJV  THE  ACTION  OF  SATURN. 

PrcVr.uimiry  Inrrstiijalhm  nf  the  Orhil  of  Sulnrn, 

Tou  tlir"  arourntr  (letcnnination  of  the  pcvtiuhatioiis  of  a  planet  it  is  essential 
tlia'c  tlie  fiinctior.s  of  the  time  which  are  suhstitiitcil  for  the  co-ordinates  of  each 
]>la  u't  in  tlie  expression  of  the  clisturhing  forces  shoultl  approximately  represent 
the  true  places  of  the  planet.  The  tlilferencc!  between  the  true  place  an<l  that 
implicitly  assumed  in  the  investigation  shoidd  be  so  small  and  of  such  a  character 
that,  when  multiplied  by  tlie  mass  of  the  disturbing  planet,  and  by  the  factors 
introduced  by  the  process  of  inti-gration,  the  result  shall  be  insensible.  If  one  of 
these  factors  is  so  large  as  to  make  a  perturbatiim  of  an  order  of  magnitude  api)roxi- 
matiug  tluit  of  the  inetpiality  which  gives  rise  to  it,  it  will  represent  an  inecpialify 
of  very  long  period  in  the  elements,  which,  though  apparently  sensible,  may  be 
neglected  for  a  great  length  of  time. 

The  pertiirbations  liitherto  found  have  been  computed  on  tin  hypothesis  that 
the  disturbing  action  of  Saturn  on  Uranus  is  the  same  s  if  both  planets  moved  in 
the  elliptic  orbits  corresponding  to  the  adopted  elements.  "We  luue  given  formulic 
for  the  computation  of  the  corrected  perturbations  when,  to  the  co-ordinates  of  the 
two  planets  corresponding  to  the  adopted  ellipse,  wc  add  corrections  represented 
by  h',  5r',  fip,  etc.  These  corrections  are  now  to  be  taken  of  such  magnitude  that 
when  thus  added  they  shall  very  nearly  represent  the  actual  motions  of  the  planets. 

Generally,  it  is  considered  sufficient  to  take  for  these  corrections  the  pertiirba- 
tions  of  the  first  order.  But  this  presupposes  that  the  elliptic  elements  are  nearly 
correct,  which  does  not  hold  true  in  the  case  of  the  old  elements  of  the  outer 
planets.  Uouvard's  Tables  of  Saturn,  the  elements  of  which  have  been  adopted, 
arc  subject  to  recurring  errors  aint)unting  to  ;}()"  or  more.  Moreover,  when  we 
siibstitute  the  new  and  more  accurate  perturbations  for  the  old  and  imperfect  (mes 
adopted  in  the  tables,  the  chances  are  that  the  errors  will  be  increased.  Desirin^ 
that  the  theory  shall  be  as  far  >is  possible  free  from  doubt,  we  begin  with  a  j)rc- 
liminary  investigation  of  the  orbit  of  Saturn,  the  design  of  which  will  be  to  give 
the  co-ordinates  of  that  body  in  terms  of  the  time  with  sufficient  certainty  and 
accuracy  to  serve  for  compiiting  the  perturbations  both  of  Jupiter  and  I'ranus, 
As  usual,  the  first  ste])  in  thi*  investigation  will  be  the  determinations  of  the  per- 
turbations of  the  planet. 


•>» 


M   9 


THE  ORBIT  OP  URANUS. 


w 


Oeneral  Peiinrhationa  of  Saturn. 
TIio  perturbations  produced  by  Jupiter  will  be  taken  from  the  exhaustive  prize 
memoir  of  llunsen.*     As  the  perturbations  required  are  those  of  tlie  co-ordinates, 
it  will  be  necessary  to  transform  those  of  Hansen  into  tlie  usual  form.     Hansen 
gives  the  true  anomaly  v  in  the  form 

V  =  tj  -\-  nh  -\-  «?i  sin  (</  -|-  mc^z)  -f-  "i  si»  2  (</  -\-  nh)  -\-  etc., 

f-nfj,  etc.,  bcin<;  the  coefficients  of  the  multiples  of  the  mean  anomaly  in  the  usual 
development  of  the  elliptic  true  anomaly.     Whence,  neglecting  the  second  power 

of  Mi'z, 

hv  =  »t!;;  (1  +  Pi  cos  g  -\-  2ej  ros  2(/  -}-  etc.). 

To  make  the  development  sufficiently  rigorous  it  is  only  necessary  to  increase  <]  by 
I  nhz  in  this  expression.     In  the  same  way,  we  have  for  the  perturbutions  of  log  r, 

hf  =  ^p„  -j-  w'z  («'"  sin  g  -\-  c'-'  sin  2 «/  -{-  etc.) 

^Pj  being  Hansen's  perturbation,  and  c'"'  the  negative  coefficient  of  cos  iij  in  the 
development  of  the  elliptic  log  r. 

Hansen  having  adopted  {otTr.ii  ""  t''*^  mass  of  Jupiter,  it  will  be  necessary  to 
multiply  his  perturbations  by  l.O'iUi  to  reduce  them  to  Dessel's  mass.  Thus  the 
perturl)ations  by  Jupiter  hereafter  given  have  been  obtained. 

The  perturbations  by  Uranus  and  Neptune  have  been  computed  by  the  preceding 
general  method,  and  are  given  in  the  following  talde.  In  the  table  I'  is  the  mean 
longitude  of  iIm;  disturbing  planet,  Uranus  or  Neptune,  counted  from  the  perihelion 
of  Saturn.  f\^  is  the  ixuturbation  of  the  Naperian  logarithm,  in  units  of  the 
seventh  place  oi'*  decimals. 


(iKN£ll.VI>  I'EKTtaOATIU.NS  Uf  TUK  LoNOITUne  IN  ORBIT  AND  TUK  LuUAUlTIlM  UlT  THK  RAIllLS 

VECTdll  or  Satlr.n. 

Ac 

ion  of  Ursi 

III*. 

I 

f     <J 

Av'tioii  of  Nept 

Hue. 

Hill 

P 

,« 

_     -         1 

'■         <J 

I'OS 

Hill 

Hill 

cua 

cod 

Hill 

1,     0 
—1 

-1-  o.ss 

-Y  8.(iO 

+  0.92 
+  0.21 

—  12.7 

-f  146.(1 

+   7.9 
—  3.5 

1,-0 
—  1 

+0.23 
+  1.93 

—0.05 
0.00 

—  2.2 

+.39.9 

—0.2 
0.0 

2.     0 
—  1 
2 

—  0.42 

—  S.1',1 

— l:i.:{it 

—  0.17 

—  2.5:{ 

—  0.25 

1-    10.2  1   —  3.7 
—  IJti.S      -1-21.4 
— .•i!)2.5      4    '■'•♦• 

2.     0 
—  1 

—2 

-1-0.44 
—1.11 
-1.27 

+0.02 

+0.02 

0.00 

—  3.0 

—  1S.5 
—41.1 

+  0.1 

—0.4 

0.0 

3,     0 
~2 

— ;i 

f   O.Ofi 

—  i.'^ 
— lo.ci 

—  2.05 

1-   I. (12 

i-2s.| 

— 20, ss 

—  1.47 

■{      0.5 
f    10.. s 
—247. 
-  (17.5 

-1  :t9.1 
+.!:i.O 

+  4'<7. 
+  42.8 

3,-1 
-2 

—.1 

-|0.02 

1  0.1(1 

—0.10 

—0.03 

—((.04 

0.00 

+  0.2 
+  3.2 
—  3.9 

+  0.4 

+  1.3 

0.0 

4,-1 

-2 
—4 

+  O.Ofi 

—  o.r,2 
f  o.:i() 

—  0.2ti 

4-  0.05 
4-   O.MO 
+   0.7:1 

f  o.o;j 

—  0.1 

—  10.5 
+    10  5 

—  10.fi 

+  O.fi 

—  12.5 
—23.4 

—  0.8 

'  UnfersiichiinifcM  iilwr  die  gegi'iiseitigon  StoruiiRPn  di's  JiipitcrH  uiid  Siitiini.s.    Von  P.  A.  Hon- 
sen.     Berlin,  1831. 

9        April.  I87S. 


66 


THE   Oil  BIT  OF    URANUS. 


I  have  Hiibniitted  tli(<s«;  perturbations  to  such  duplicate  rompiitiitions  niul  otliri 
checks  us  lead  ine  to  believe  that  none  of  the  terms  can  be  in  error  by  more  than 
a  small  fraction  of  n  second,  but,  us  they  arc  not  intended  to  form  the  basis  of  a 
definitive  theory  of  Saturn,  I  do  not  vouch  for  their  absolute  precision. 

In  this  provisional  correction  of  the  orbit  of  Sutuni  only  heliocentric  longitudes 
liave  been  employed.  These  were  ilerived  fttr  a  series  of  dates  from  Airy's  rechic- 
ti«m  of  the  Ureenwich  observations,  the  modern  (Jrcenwich  observations,  and  the 
Washington  observations. 

For  these  dates  the  value  of  n^z  for  Saturn  wns  computed  from  the  formulir 
found  on  papes  189  and  lJ)()of  the  work  of  Hansen,  already  quoted,  omittin;;  all 
terms  less  than  1",  and  including  only  tenths  of  seconds  in  the  results.  The  dates, 
the  resulting  values  of  nh,  of  the  factor  <-,  cos  («/  -|-  J  iifz)  -f-  'ie-i  cos  2  (y  -|-  |  uJz), 
and  of  the  concluded  tv  are  as  follows.     The  formuhe  for  Iv  is 

1  .()'.»1(>  »»'3  j  1  -f  e.  cos (i/  +  J  ii'z)  -f  •ic, cos 2 (i/  +  J  nh) j . 


,«r=1.021(> 

n'z 

Pfttn 

Or.  M 

■uii  Nuoii. 

17  Jl 

May 

:n 

n-)7 

Aug. 

7 

17.VS 

Aug. 

27 

17(51 

Oct. 

G 

17();l 

Nov. 

1 

17(!.> 

Nov. 

2:{ 

177:$ 

Feb. 

2() 

1780 

May 

24 

17!)4 

Nov. 

k; 

180'> 

Feb. 

2:{ 

is-.»;j 

Nov. 

l:{ 

I8;ii 

F.'b. 

18 

18:}8 

May 

1!) 

184.> 

Aug. 

17 

l8.-,i 

Nov. 

l.» 

1S()() 

F.b. 

14 

18G7 

May 

l.j 

nd 


Fiu'(or. 


m 

—  1792.7 
— 20:i*<.2 
-215;j.2 

—  2(»(54.5 
-:no7.l 
— ;{r)()4.() 
— .'ir).)7.2 

—2(540.7 

-;i7:n.!) 
— :M2').r> 
— ;J0:J(5.8 
-;5(571.7 
-2777.9 
—2220.9 
— :{l."»9.:l 

— ;j4(58.;i 

—  2227.G 

The  perturbations  by  Uranus  and  Neptiuie  were  computetl  from  the  values  of 
their  terms  just  given.  The  principal  terms,  the  sum  of  which  make  up  tlie  helio- 
centric longitude  resulting  from  the  adopted  elements,  are  shown  in  the  first  of  the 
following  tables. 

In  the  next  table  we  have  after  the  date  the  heliocentric  longitude  from  Bouvard's 
Tables,  as  deduced  from  the  longitudes  given  in  Airy's  reductions  of  the  tireeu- 
wich  Observations,  from  the  Anhonomi'm/ioi  Ju/nlnwh  for  18;ll,  and  from  the 
Nautical  Almanac.  Then  follow  the  corrections,  roiighly  deduced  from  ob.serva- 
tions  made  near  the  opposition.     Adding  these  columns,  wc  have  the  longitude 


1947.7 

—.0990 

21;J4.2 

— .()Go2 

2212.') 

—  .0474 

2"»4(5.-> 

+.0244 

2880.;} 

-f.0729 

;j()9").l 

-{-.1(>82 

;j;J42.o 

+  .0419 

28.»S.2 

— .09->G 

:W21.I 

+  .1017 

:51S4.7 

+.0529 

27l(5.;J 

+  .0944 

;w78.-i 

+  .0(5:59 

297(5.7 

—  .08(5(5 

2:J42.7 

—  .0721 

2847.0 

+  .08(5;$ 

:}l(il.4 

+  .0740 

2;n:5.i 

—  .081*i 

,. 


Tin:  ouiJiT  OK  uiiAXua. 


•7 


from  ubserviitiuu.     To  the  ri^^'lit  of  thoHu  uro  tlie  equations  of  condition  for  llic 
correction  of  tiic  dcnu'iits. 


IWlurbttioim  lijr 

I.OIIK.  of 
I'lirilivUou. 

M.'i>n 
■ijiiiualjr. 

Ei|nali<in  of 
uviitra. 

R«il.  to 
Kcllptio. 

Nuts- 
tlou. 

TruB 
lougitudo. 

Nwp- 
tuu«. 

II 

Jupiter. 
/        II 

UrtDui. 

II 

O          1           II 

O        »            II 

0        /          1' 

/ 

II 

O           1          II 

8S  41  27.5 

150  55  50.1 

-1-2    3  5(1.  fi 

—20  52.7  ■ 

-45.7 

—2.7 

+  1 

»n.9 

+  15.8 

250  12  25.8 

SS  4<>  41.2 

2311  13    7.1 

—5    7  38. 1 

—33  58,2 

— 3(l.(i 

+  1.0 

—  1 

20.5 

—11.8 

310  Hi    4.1 

H.S  47  :il.4 

248  25  53.1 

—ft  48  30.4 

—35  53.2  1 

-35.2 

+  17 

— 1 

35.8 

—15.0 

330  4(1  30.(1 

8S  50    7.« 

28(1  211  20.8 

— ti  Itl     1.3 

—44  24.(1 

-  3.3 

+  2.0 

— 0  43.5 

—  14.1 

8  15  13.5 

8H  51  51  « 

311  44  13. ti 

—5    0  5S.2 

—52  37.1 

f  11.0 

+  2.4 

+  0  43.3 

—  4.9 

34  43  22.0 

Hs5:t;j.j.l    :i;if,  jV)  aii.i    _2  41  lo.ii 

— i58  24.0  , 

f  10.2 

+  1.0 

+  1 

3tl.(i     +  (1.7 

tl2  12     1.0 

8H  5fl  :«'.».« 

(15  40    2.3     -l-t!    0  35.0 

-50  17.2  - 

-32.0 

—0.8 

—  1 

37.1 

+   4.8 

150  38  54.0 

8!>    5  4:i.4 

154     8    4.8 

+  2  37  57.7 

—44    0.7  i- 

-53.8 

—1.7 

+  1 

37.5 

—14.5 

245    8  13.2 

8!»  17  50. 7 

331     11  10.4 

—3  18    5.2 

— tI2  17.0   +   0.0 

+  3.5 

+  1 

30.0 

—  14.1 

50    5    7.9 

89  2:»  55.7 

50  5tl  17.0 

+5  44  4tl.fi 

_57    5.5  — l8,tl 

+  2.5 

—  1 

37.3 

+   1.4 

154    0    2.7 

8!)  42    7.0 

325  22  2tl.O 

—3  51  58.0 

—50  3(1.8  -27.0 

—3.0 

1-1 

21.(1 

+  14.7 

50  23    3.0 

8!»  4H  11. « 

54  10  33.1 

+5  25    2.(1 

—til  11.7    +13.3 

+2.5 

—  1 

33.2 

—  7.4 

148  21  11.1 

8!»54  ir.i 

142  44  37.0 

-t-3  40  38  1 

—4(1  17.0  —  3.4 

+  2.1 

+  1 

20.7    —  4.3 

235  34  37.4 

90    0  20.4 

231   18  40.0 

—4  47  3ti.tl 

—37     0.0   - 

-35.5 

+0.0 

11.8     +14.8 

315  52  52.2 

no    •>  24. ti 

310  52  44.8    _4  21  53.0 

—52  30.3 

+  4.7 

—0.0 

+  1 

8.8     —18.4 

44  45  31.3 

00  1 2  2H.  7 

48  2ii  48.7     +5     1  4tl.4 

—57  48,3 

f30.3 

—3.4 

— 1 

25.1     +14.1 

142  42  31.4 

•JO  18  32.8 

137    0  52.«     +4    8  31.2 

I 

—37    7.(1  - 

1 

—  8.2 

+  1.3 

+  1 

21.(1    —  4.1 

230  52  50.7 

Dat«. 

Tal.iilir 
hiiigitU'la. 

Otis. 

cor. 

I.OI1H.  from 
ubixtrvntiuu. 

KquATioNs  or  Condition. 

1 

1751     Mav    31     250  13  38,0 

II 

—    8.5 

250  13  20.5 

(!3.7  =  +0.fl()«, 

—44*11  +0.nO««'  +1.82»'«u 

1757    AiiJt,  27     310  17     8.5 

—  is  (1 

310  It!  50.5 

4(1.4          0.0  4 

—30      —1.53      -fl.l4 

1758    Aiitf.  27     3.10  47  37.0 

—  11.3 

330  47  23. ti 

47.0          O.Oti 

_40      — 1.7t!      +0.80 

17<11    Ocl.      C. 

8  15  13. S 

+  0.2 

8  15  14.0 

0.5           1.03 

—30      —1.00      -0.50* 

17t;3    No?.      1 

34  4.!  42.3 

-f-15.3 

34  43  57.(1 

35.0           l.tl7 

—30      —1.(13      —1.31 

17(15    X(»v.   2.1 

tI2  12  2t».7 

+  211. 1 

(12  12  41   1 

40.1            1.11 

— ;iH       —0.88       — 1.88 

1773    Ffl>.    211 

15 It  40     3.8 

+  IS.S 

150  4(t  22.(1 

HS.((               I.(l( 

—28       +1.03      —0.70 

1780    .M:iy    24 

245    0  17.tl 

+  T.8 

245     0  25.4 

72.2          OOtt 

—  I.s      +0,7(1      +1,74 

I704    Nov.   Hi 

5ll    5  38.8 

fl4.7 

5tl    5  53.5 

45.  tl           1.10 

—   ti       —1.08       —1,78 

lsM2    Ki'l).    23 

154    7    2.7 

i  Kt.s 

154     7  13.5 

70.8           l,tl5 

^,     2        f  1,85      —0.08 

H-23    Xov.   13 

50  23  31.5 

+  21.3 

5tl  23  55. M 

.52.2           l.tiO 

+  2ti      —1.27      — l.titi 

1S31    K.l).    is 

US  22  3S.5 

f-   3.tl 

MS  22  41.5 

Otl.4           l.tl7 

t  33       +1.75      — l.ltl 

1S38    .MttV     lit 

235  WW  11.2 

+   3.4 

235  .1(1  14.t; 

O;  a           0.01 

}  .35        1  1.08      -1  l,5tl 

1845    Aii'tf.   17 

315  53  42.2 

fl5.7 

315  53  57.0 

(15.7           0.03 

+  42      —1,43      +1,28 

1852    Nov.    15 

44  411  15.4 

+   5.0 

44  4(1  21.3 

50.  t)           1.00 

1  57       —1,42      —1,53 

IXCO    Kcl).    14 

142  H  111.2 

—  13.7 

1  12  44     2.5 

01.1           i.tts 

1  (;5        t  I,tl3      —1,35 

18)17    Miiy    15 

230  54  17  s      -1    r,.l 

1 

2.10  54  52.0 

113.2       +0.02 

+  (12       ^  1,2,3       (  1,45 

A  normal  equotion  for  iV  is  ohtiiinod  hy  tnkiii}'  tlic  sum  of  all  the  rqnntions. 
Tliiit  for  hi  is  formed  by  8ul)triirtinj;  the  sum  of  the  first  seviMi  from  the  sum  of 
the  last  seven,  and  those  for  <V'  and  e\i  liy  taking  the  sum  of  the  equations  in 
whiih  the  coefficients  of  rV  or  e\i  are  jjreater  than  unity,  after  chanpin<;  the  signs 
of  tiie  eciuations  in  whidi  tliey  are  negative.     The  normals  thus  obtained  arc 


1' 


(k) 


TIIK   OKU  IT   OF    I-  IIA  N  U  S. 


0.04  -i-5H7 
■  2.14  -f'207 
-  2.11      —  GO 


These  equations  givo 


—  'i.Ki.V 
-f  G.m;} 
-1-21.58 

-f  a.Gi 


-  3.irv.\j 

—  o.oy 

+  2.10 
-flO.GG 


-f  lOGO.Jl 
-f  230.;} 
-H  208.1 
-I-     63.8 


if  =  -f  64.8  (Epoch,  1800.) 
An  =  4-    0.2G8 
ie  =-]-  12.6 
c\j  =  4-    8.2 


— Substituting  these  viilues  in  tl)o  seventeen  equations  of  eondition  we  have  the 
following  residuals,  or  e.\ce8ses  of  theoretical  over  observed  longitudes: 


1 

-f    7.2 

10 

-1-10.7 

3 

—    4.3 

11 

—    G.4 

3 

-  10.2 

12 

—    1.1 

4 

+  21.7 

13 

—    0.4 

6 

—    9.0 

14 

0.0 

6 

—    7.1 

15 

+    3.7 

7 

—  11.3 

16 

+    4.1 

8 

-f    8.2 

17 

—    7.7 

9 

—    6.4 

These  residuals  are  much  larger  than  they  sliould  be,  and  I  scarcely  know  to 
what  cause  to  attribute  their  magnitude.  The  results  are  however  amply  reliable 
for  the  purposes  of  the  investigation,  and  lead  to  the  following  elements  of  Saturn: 


n, 

90     6     26 

h 

14  r,{)      3.2 

e, 

112  20      0 

«. 

2  29     39.2 

fi, 

4399().395 

e. 

.O'jGOGGO 

log  («  -f  fta),    0.n7i)(i7G 

Kpoch, 

1850,  Jan.  0,  Clreenwich  mean  noon. 

It  will  be  seen  that  the  adopted  position  of  the  plane  of  Saturn's  orbit  is  retained. 
It  was  corrected  from  observations  before  the  perturbations  were  finally  computed. 

Of  the  above  corrections,  those  of  the  epoch  and  mean  motion  need  not  be  taken 
account  of  in  the  corrections  of  the  co-ordinates,  since  the  mean  longitude  remains 
in  the  formultc  as  an  arbitrary  quantity  to  the  end.  The  effect  of  the  correction 
of  the  mean  distance  is  insensible.  The  corrections  of  eccentricity  and  perihelion 
are  therefore  alone  to  be  retained.  They  are  allowed  for  by  adding  to  Iv  and  iif 
the  terms 


I 


u^ 


irisrwS5!U!a»ir«£v",«jaia*aiWlSS*aB«ftiKia^ 


T  UK   ORBIT     >F   U  KAN  US.  W 

fiv  =       2^e  sin  </  —  efiu  cos  g 
=  -f-  'i^'-'i  si"  J/  —  1(»'.4  tos(/; 

ip  =  —  ^e  cos  </  —  c^u  sin  // 
=  —  ll2".6  cos »/  —  a". 2  cos  </. 

rbrturhaliona  of  Saturn  and  Uranus. 

Tho  following  oxprcssioiis  include,  with  tlu'st<  corrections,  all  the  perturbations  of 
Saturn  and  Urai  us  which  can  produce  any  apjircciublo  perturbations  of  the  second 
ord(  in  their  mutual  action.  In  these  expressions  the  initial  letter  of  each  planet 
is  put  for  its  mean  longitude  counted  from  the  perihelion  of  Uranus. 


I 


rEllTUBIIATIONS  (ir  SaTIB.N. 

Argiiiueiit. 

»o' 

1 

■in 

fOH 

IM>« 

Hill 

» 

" 

tt 

II 

S 

+  2.').  2 

f   21.4 

—    2.7 

+     0.0 

.S 

+      l.ti4< 

-      1.4H< 

—     0.H0< 

—     0.74/ 

S  —   J 

—  2<l.l 

+    fi.a 

+  17fi.7 

_  1.5.0 

iS  —   J 

— 42l.ti 

—      8.8 

—  ll.'i.s 

—     9.1 

as  —  J 

—  .'.4.7 

—  12.4 

+    12.3 

—  lfi.fi 

2.S  —2.; 

+  :t2.« 

—     1.0 

+  2H..5 

—     1.1 

:i.s  —2,7 

+  27.7 

—  21.3 

—  l.-i.l 

_  12.3 

4S  —2,/ 

f  2i(;.9 

— 7(i«.a 

+   11.4 

+  403.0 

6.S  —2./ 

+2:iita. 

— icita. 

—  .'■>8.8 

—  34.4 

CS  —  2./ 

+  1J3.0 

—  !l3.f. 

—  M.\ 

+  f)fi.!» 

rr-  s 

-f      H,« 

f     0.3 

+     3.0 

_     0.1 

2ir—  s 

+     0.8 

_    s.s 

+     0.1 

+     1.4 

■2U  —  2S 

—  I.J.  4 

f      0.2 

—     8.0 

+     0.1 

:iU—  S 

—     «.4 

—  27.0 

+     O.l 

—     0.7 

3U  —  2S 

-1-   1«.3 

—  14.« 

+     8.8 

+     7.0 

I'EIITL'RBATIONS  OF  I'llANllS.                                                                                1 

Argument. 

t 

u 

1 

sin 

coa 

COS 

(in 

It 

It 

n 

11 

U 

—  n.iu 

—     0.2f« 

4    0.05< 

—  0.14< 

U—  8 

— 2(I.S 

+    s..--, 

+3tl.O 

—  4.0 

217—   8 

—  ll.!» 

+  14:i.,5 

—  2.0 

— fi3.8 

•.U7—   8 

+  411.3 

+  11.^.'.) 

—  4.7 

+  13.  a 

2(7-28 

+  4.1 

0.0 

+  4.2 

+   0,1 

a  (7  — 28 

+  2.1 

—     l.B 

+   l.S 

+    1.4 

4(1  —  28 

f  0.6 

—     1.8 

-\-  0.4 

+    1.3 

riU  —  28 

+   1.2 

-f     ;t.o 

f   0.4 

—  i.a 

Let  us  now  resume  the  equation 

'^  ih         H  i    ra 


0^> 


\,^ 


70 


Til  K   OllHIT   OF    UllA  NUS. 


Brginiiinp  witli  thr  last  two  trrins  of  tliin  cxprissioii,  it  miiy  be  sliown  at  the 
outset  that  thry  are-  (initc  iimciiNilih',  The  vtivvt  of  llie  constant  terms  in  fip  and 
ip'  iias  alr.Muiy  l)een  inehuled  by  correeting  the  lofjaritbm  of  the  mean  distanee  by 
their  amount;  they  are  therefore  omitted.  Tlie  Iar<,'e.st  remaiiiinf»  term  is  ti4',  the 
square  of  which  k  only  ()".()•.'.     la  the  proihict  ri',Y  tiic  hirjjest  terms  urc 

+  0.014 

—  O.OlDsinf/ 

—  0.01 1  sin  (%  — 20 

—  0.011  voH(4(j  —  'H) 

which  may  be  entirely  no<jlected. 

We  shall  thereforo  only  consider  in  liQ  the  terms 

rip 
As  already  reninrked,  It  is  rijjorously  a  function  only  of  T',  p,  and  p',  T  being 
the  angle  niaile  by  the  radii  vectores  of  the  two  planets.      Hut,  in  the  analytical 
development  of  /;,  the  qtuintity   V  is  considered  as  a  function  of  v,  v',  and  y,  so 
that  we  have 

^=/(p.p'.v,  v',y). 

In  the  previous  computation  of  the  perturi)ations  of  Uranus,  wc  have  supposed  li 

to  be  a  function  of  p,„  p'o,  etc.     The  corrections  to  Jl  and  its  derivatives  with  respect 

to  V  and  p  are  now  giv«>n  by  the  equations  (11),  with  the  modifications  shown  on 

pages  24  to  27.     The  derivatives  of  Jf„  which  enter  into  these  e(iuations  are  formed 

as  follows:  If,  in  the  value  of  Ji  produced  by  the  action  of  Saturn  on  Uranus,  wc 

consider  any  term  of  the  form 

in'h        -- 
cos  iV 
(I, 

where 

the  accented  (piantities  always  r(<ferring  to  Saturn,  hut  a,  being  the  corrected  mean 
distance  of  L  ranus,  then  we  shall  have  the  following  terms  in  the  derivatives  of /f. 


' 


6R 

f)It 

6R 

r9p 

8^R 

'6\' 


■■—  „    ('+.y)'*'i>  -V 
"1 

-'f  (/'  +  /)  sin  AT 
"I 


m'    fh 

„     cos  ^V 

</,     tin 

I'i'fl    ,■,-.■,  XT 

—  .,    ('+^)-cosJV 
"1 


THK   OUUIT   OF    UU  ANUS. 
dm  m'h 

iJUi 


71 


(•+;){«■'+;■)  cos  iv 

■,    =  (A+  J3    )(*  + J )»'"•" 


6vd^ 


dvfif  '' 
fit 

sm 

6/  '' 


an  _  ivii 

ov       t'Vf'yj/ 
m'  vh 


(«-|-y)siii  Y 
h 


(1,  On 

"''(,.+  ':;M(.'+y)Hi„.v 


•9'« 


c'fjcVj)' 


It    ?H    /  f'rt    j^   f'VJ  \ 


COS  AT 


All  tlio  numcrinil  data  lU'ccssary  for  the  computation  of  these  derivatives  have 
lieeii  jjiveii  in  Chapter  II.  C'oml>iiiin^  the  terms  having  the  saiiie  ar;{umeiit,  we 
fiiul  tiio  foUowiiiji  values,  omitting  those  given  in  Chapter  II,  and  tliose  which 

arc  derived  from  the  others  by  mere  addition.     The  terms  of  '    .,    are  also  omit- 


fVV 


ir-n 


ted,  because  they  are  sensibly  the  same  with  those  of  \   ,  ,,  changing  the  algc- 
'  '  t/Vc/V 

braic  sign. 


0, 

dlt 

0| 

rru 

a, 

«9'«                1 

m' 

t*v' 

wi'  c^v^v' 

«•'  (5('(9v'               1 

9      I' 

sin 

CIIH 

sin 

COB 

Hill 

COS 

0,-1 
—2 

+0.2^*74 
-fo.oonii 

0 

—0.0310 

0 
+  0.0310 

+  0.2x72 
+0.00(i7 

+0.2(101 
+0.00(10 

0 
—0.0322 

1,     0 
'2 

— ;j 

—0.0103 
— :i.4Mii 

— O.OiK'.M 
+  0.03S8 

—0..     11 
—O.lit.lO 
+0.4.i04 
+0.0173 

+  0.0100 
—0.0021 
—0.4304 
—0.0179 

—0.0101 
— 3.4S10 
—0.1039 
+0.03N9 

+0.0249 
_5.2(1S4 
—0.0703 
+0.0308 

+0.1104 

0 
+  0.45sn 
+0.01(13 

—3 
—3 

—0.001.') 
—0.0.544 
+0.4104 
—0.0053 

—0.0035 
—0.0713 
+  0.0120 
—0.01  111 

+  0.0021 
+  0. 142(; 
—0.0171 
-f  0.0300 

—O.OOHO 
—O.OCSO 
+  0.S233 
— 0.022(» 

+  0.01,55 
—0.17(10 
— 1,(515 
+0(I2HH 

+  0. 00f.fi 
+0.2403 
+0.0014 
+0.0547 

3,_I 
—3 
—3 

—0.0070 
+0.0670 
+0.2542 

—0.0003 

— o.o(;5o 

0 

+0.0171 

4  0.lo,'-.7 

—0.00(15 

—0.0210 
-t  0.1013 
+  0.7(124 

+0(1107 
—0.1701 
—  1.0S84 

+  0.0304 
-I-0.2S40 
—0.0071 

4,-2 
—3 
—4 

— O.OOH 
+0.0515 
+  0.1448 

—0.01.10 
— O.Oi'.tO 
—0.0015 

+o.o;tfis 

+  0.1 0H7 
— O.OOKi 

—0.0103 
+  0.14.50 
+O.57K0 

+  0.0213 
—0.2144 
—0.7(144 

+0.0547 
+0.2(147 
—0.0005 

78 


TIIK   OUWIT   OF   UK  ANUS. 


«i 

oJi 

n. 

ffit 

«. 

irn            1 

<J  I 

m' 

d9 

in    i)\df' 

"«     ritdf                   1 

Bill 

I'OH 

«iii 

con 

Hill 

Ct.i 

0,     0 

—1 

—0.02(1 

40.172 

— 0.5(12 

4'o.r.'5n 

40.002 

U 

40.113 

-0.04.') 
—0.(110 

1,      0 

—a 

-l-o.oto 
— o.oo;{ 

-fO.HS'J 

-}  (lOlf) 

48.741) 
4  O.lHO 

40.01-1 

— 8.7.'>0 
—0.17(1 

40.070 
— o.col 

40.888 

—0,227 

40.011 
-j-o.aia 

—0.07  ft 
-i  B.I8» 
40.100 

2,-1 

—a 

40.084 
40.018 

-(  0.248 
40.472 

— 0.2:10 

—0.04; 

40.100 
40.013 

— o,:i22 

—0.00(1 

40.020 
—  l.OS.'i 

•'1,-1 
—a 

40.013 
4  o.oTil 

0 

40.oo< 

4o.O(>8 
40.278 

40.01;' 
—0.122 
— o.n;i4 

40.n'Jl 
-1  0.21  ft 
—0.007 

-0.049 
—0.344 
4  0.010 

40.028 
—0.244 
—1.2(10 

4,-2     ' 

40  oir) 
40.001 

—0  001 
-l-O.li.ft 

40.020 
-  0  111 'J 

40.042 

—0.071 
40.001 

4  0.01!) 

— 0.8f)4 

Till!  iloiivations  witli  n'spcct  to  y  iiixl  tl)(>  node  \\\\\v  hrcu  omitted  'ircniiso  tlicy 
lire  (jiiite  iK^f'nsihle.  Tin-  terms  of  hii  (lepeniliii;;  on  tlies<'  deriviitives  iire  f,'iveii 
liy  equation  (;J1>.  In  tlie  ease  of  rramis  diMturlu!!  by  Saturn  the  largest  values 
of  the  coelficient» 

4,/Moti^;  i/.tau  Jy,  J  '^^^■ 

are  only  ahotit  S)h,  .vliile  the  larjjest  roefficieiits  in  hh,  hk\  and  }y  are  less  than  10". 
l^''ne(>  the  lai'p>st  terms  in  (31)  \viil  lie  >  tlu  order  of  niiigiiitiide  {)".h  miiltiplied 
•>  ■  the  mass  of  Saturn,  and  may  therefore  lie  omitted  entirely.     Omitting  them 

the  values  of  hit,  h    .        »niJ   ^    .,     become 
w  tip 


iltr^ 


6R 


^r -f 


rn 


h'  4-    .      ho  -4 


PR  .     ,    f  R  ,  ,  ,    o'R 


i'u  ,     ,    f  It  .  ,  .    in  J      . 


dv 


'R 


(Vt'V 
trR 


R 


r-R 

{R 


,V' 


=  .     ^W  4-  ,    ,,     hv'  -\-     .    .    hi)   -\     .     .    ,ho' 


» 


All  the  separate  faetors  frmii  which  the  second  members  of  lhes(  e(|natiniH 
are  formed  have  alreiuiy  been  given,  lorming  their  |uodiu  ts  in  tiie  way  described 
in  Chapter  11,  we  have  the  res  ilt  f:  ven  in  tlie  following  tallies. 

The  expressions  for  hR  are  a;r;ii  ged  so  that  the  vi.Iiie  of  P',hR  can  be  olitained 
(rom  them  by  direct  diflereiitiat'on.  This  is  done  by  disfingiiisliing  the  time 
inti'odiieed  into  R  by  the  co-ord  nates  of  I'ranus  from  that  iiitroilticeil  by  the 
co-orilinates  of  Saturn. 


TUE  OUBIT 

OF   URANUS. 

78 

2",  (fV/ 

...•+^»vj 

u  s 

COS 

tf 

U 

S    J 

II 

If 

'.     0 

—  I.40< 

—  i.3a< 

+  1. 

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74 


TIIK   ORBIT  OF   U  KAN  US. 


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--  tt'iU 

)     l.4t( 

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—  o.r,o< 


—  45 

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T  II  K  ORBIT   0¥   U  K  A  N  U  S. 


76 


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70 


TIIK   Oil  HIT  OV    UllA  NHS. 


Fii    the   trrnis   of   hit   intrtMliiccd    liy    ilu;    pcrliirlmtions   of    Sahim,    niimcly, 

^  ,  it/  -^-      ,  Aj)',  tlic  tlifri'rcutiatioii  r<'|ircH«'nt(><l  liy  />*,  Klumltl  he  |M>i'toriii(>(l  liy  con- 

RidcriiiK  At/  iiiul  fif)  as  t-oiistant,  altlit)U(r1i  tlicy  arc  <-\|ir(>NS('(l  art  n  fiiiu-tioii  of  tlio 
iiu'un  luii^ittiilc  of  l!raiiiiM,  as  wi-ll  uh  o(  Saturn.  'I'lic  mean  loii;;itiiiii'  of  I'rantw 
thus  iutrodui'cii  is  tliinfori'  rcprcMcutcd  hy  /",  which  ia  regarded  uh  constant  iu' 
taking  //,/?,  and  //^only  supiMwd  to  vary. 

Again,  iu  the  teriiis  "  '  jf4-        ^"•••''kc  I'r  and  A/i  reiireHent  perturhations  of 
t^'v         '    5p    '  ' 

Iratnis,  their  roinplete  derivatives,  with  resjXM  t  to  the  time,  i:r<>  to  he  taiveu.      Ihit 

their  exprexiiiuDs  eontuin   tlie  mean  lon^'itiuh- of  Saturn  us  well  as  rrunus.     'llu? 

mean  hmgitudc!  of  Saturn  thiiK  introdiieeii  is  representiMl  hy  .S*.  and  iK  to  he  eon- 

fiidered  varialde  in  ohtaining  iy,hll,  while  S  is  eonsidi  red  eonstaat.     The  ratio  of 

the  roeffieient  of  t  to  it   in  tln^  varioUH  terms  of  thin  part,  of  ftit  is  given  to  the 

riglit  of  eaeli  rorres|N)tiding  ti  rni. 

'I'lie   vahie   of   //,A/i'   iM'ing  onee  ohtained,  there    is   no  hiUfjiT  any  disliiutinii 

necessary  hetween  /',  V\  or  l>etween  iS' and  .S".     'Ihe  himdar  terms  are  liu-refore 

eomltined  hy  jtiilfing  .S"       .V;  /"       r. 

Ironi  llie  ul»<»ve  vahies  of  'J:!)J))  U  and  'i,S  '      ,     we  hirni  the  following  value  of 

'V 

"'  H'-  '^"  J' A/>'. /•'/'+   "'   ^V~ 

and  of  the  other  fjuantities  whiili  enter  th(  |»er!urhatit»us  «d"  the  ro-ordinates.  We 
liliall  l>e;;in  with  thos<>  lernis  wliirli  depetid  mdy  on  th<-  nnihial  action  of  Saturn 
and  I'ranns,  lieeansc  tiie)  are  (i  w  and  small,  and  tiu'  only  terms  whit  h  are  sl■n^illle 

are  those  in  whiih  the  rocHMi  nt  of  tin-  mean  liinj,'itiiile  of  Saturn   is   I.      \\e 

shall  therefore  (online  oursi'hes  to  these.  And,  instead  of  enntlii)inj,'  the  eoii- 
iliiised  furmnhe,  Wf  shall  make  the  en;!i|)ntation  in  fnll  liy  (l-l). 


".^Q 

II 

{AV 

"',<!*« 

in 

in 

CIIK 

hIii 

i.i 

In 

HI* 

.iu 

1 

i  tc. 

t'          n 

„      „ 

n         II 

II        II 

II 

It 

*/         II 

).    1 

1     (7                  (    7S 

III 

t  •*"< 

1      T 

1  i;i:< 

0 

it+o..'.;w  i  +  ;i<it    (i.M/ 

4-    l:i4i)  *>-.i' 

f    4'.l-f  (1  KVM 

1    :i.i... 

■O.dC'/ 

-      S-f(Mll.' 

~  L'l  )  a.iei/  +  (,-.'. -siic**  1-    II  f  .' :;7' 

I  |:is  ,  I  K-H 

S    '-'It- 

-1  s,">/ 

-If.l  i  •-'  llr 

•J 

•  -  fid  1  i  -ni  —  2s  1  :t.su  \— ■>',[)     \*\M 

t  t.il     :i  (i.W 

)  -.'.'ill- 

(1  i.!( 

4-    K.'.— -J  711/ 

■  :<•>:>  (  0  ;il;  ■  t  ;.'|ti-_.i.v.i< 

1       7  1  -i-WM 

:;i  t  •.'..■!0< 

—  ;i(i  ( 

•i  1  \t 

1     14— 1  ti-J/ 

* 

—•iwx  1  a  l,>< 

,  IJii     -'  IU 

~'X\\'X  ) 

II  l.'i/ 

--l-'(»  )  "J  741 

i>    '■ 

r>is                 1                 Hill 

•» 

am              1 

foi 

//             n                    '            tt             II 

"           '"          1 

0—1 

{  aoi-    oiuHtii        —  o>i:i  i  imon-.v 

1  a  oii'j  )  a  (Midi)/  1 

\  0  (!'.>  1    (  0  iKIrtdf 

1, 

t  nil  ,  »(Mi:i:(#        —  (CK;  t  """i:i:t> 

1  M  lilt  I  ii.aaiii,/ 

1  0  (i.'is  t  a  iiihMi/ 

o 

♦  :i:;;i  (  imuiti         _  iw  (  mi.iii 

1  IMV.'a  \  (>,(IO!f'l 

(  0  so:t_.(i  (ian7i 

•'l, 

__  1  ir>_  (itin(i-j<         ♦  II71I     Oi'ir."* 

f  >.'(',!)        (1  (Mil  W   1 

4 II  s;tu— (1  iii.Mu 

4, 

_.  an                        i.iii'i 

|ti  l;t'j            11/ 

1,  i>  uta  1 II  iMiiiii/ 

TlIK   OUniT   OF   URANUS, 


IT 


The  rompiitalitiM  of  tli<w  tcrntH  hciiifj  oxtrrmcly  roiindrx,  t\  chcrk  njKin  tliiir 

ncciirary  is  (Icsiru'ilc.      In  llic  '.  asc  of  tlic  secular  viiriiilit>iis  ni'  llic  (■((••Hiciciiis,  ilio 

nil  (Ui'ii'iits  of  the  tiiiK"  ar<'  easily  oMaiunl   liy  n-il).stinititi;^   in  tlic   inlef;mt«'ti  per- 

tiirltatiiMM  tl»o  vuriwtioui  of  tlie  et;n  iilrieity  uml  iKTiliclioa  of  Suturii.     Thus  I 

htive  fuuiitl 

»  n 

fv  :=  I  0.i)IO;J /  sin  ('i,j  -  t)  -  ().()(»•>*  t «os (-2;/  —  /) 
4-0,0027  I  sin  (>ij  —  I)  —  O.Oiaw  t  io»  {.iy  —  /) 

Tlie  jjfrefifi'Ht  iliserepitury  is  AmiikI  in  i1m>  coeffii  ient  of  sin  {i)ij  —  t),  mnl  it  anionnts 
to  (I  ((();|^/,  or  iiliont  U"  I  in  a  lentiiry,  IJutj  owinj^  to  the  ;;re,it  |Mr;o(i  of  this 
term,  nearly  (iOO  years,  tliis  dift'erenee,  thirin^r  imy  one  centurj,  v  ill  lie  nearly 
t'liniinateil  llH-on<^'h  ihe  mean  !(>U|U'iliiil<    ami  nnan  tiiotion. 

It  may  aUo  hi  rei)iark<il  tlial  in  tl>i'<  ruse  the  terms  derived  ftom  the  pertiir- 
bations  of  tlio  elements  are  imdouhtidly  thy  correet  ones,  and  will  therefore  ho 
employed. 

'I'he  fi>rins  whiel,  tin'  pveci-diii^  iiite^nition  fail-i  to  give,  owin;^  to  the  constant 
t< mis  iiUTodnc(  d  into  ^ls(^  and  yii\(^,  an   I'uund  l>y  {'22). 

We  tiiiiH  have 

w^; /),,;.;;'  r-  .x.  (»■■.;«» 
«:;■/,./.•;'=- -fo. '27 

r,».^p  1=  I  ,)[„!'  jtr.HC  sin  ,j  -   i)".-'!  <os ,/{ 
.Sji  =  J  J/,»/' ;(j".;i(;,,)s./  |  ((■.•J7r,in//i 

~-  r-  |i!'.()()tH((>;iS((»s;/   f  (r.il(l()(K».".)sin</j. 

'Ihe  (i^rialest  etreet  of  thi'se  terms  anioimls  to  less  than  tnu -twentieth  of  a 
fieennd  in  a  ceiitiny.  Tin  y  may  Iht  rctore  he  inijic  eied  in  tin  present  theory. 
The  other  teiiiis  lOiilaininf^  the  MCjiiave  o('  tile  tiiiie  are  yet.  smaller. 

Apphiii;;  the  terms  of  the  -.eeorld  order  tlilis  t'lUnid  to  the  terms  of  tile  liis(  order 
di'pendini;  on  the  iurre>p<)i'.d<ii}{  ar^nnieiits,  the  p(  rturhatiou.sof  I  ramin  h}  Saturn 
become 


i 

it 

r 

+«, 

:;     '■ 

KID 

f« 

CUK 

u 

Kit* 

niii 

-1,    ' 

\   « ti  trt 

1            11    II^I'.I 

f     0 

_   a 

ly     1 

,     1  Jsj 

,     0  i  ;•• 

-.  II 

—  .11 

1.  -1 

-^.HITM    » 

nmr 

■J,       K  .,M\ 

t  I7.'):t 

—•-'11 

J       1                      II  ,!7H    1 

1  .»ir 

\\M  -.'i...  — 

11.04  7' 

-  S7 

— .•im 

:;      1                f.'.l  !»!!     t 

0  j;  T 

\\\f<  r.!i    — 

1  its  T 

—2.17 

t.iii 

II                I    'i-Mt 

(     .'.  f.r.c, 

—  .'..-i 

( till 

;i      1               -+   »  i;:il 

+    o.:ja» 

—     4 

-f    10 

A 


T  here  I'  j'r"se,ii-t  tli.    liiii.-  counted  in  crninries  fvipin  1^."<<I.(I. 
The  utber  teriM>i  remain  the  same  as  f;iven  on  pi;,'e  ■'")(). 

/V,  'iirhiiiiiiiiM  i/ijM'iniini/  oil  III'  jirinliit  I  nf  f/i,   imti-  s  nf  Jnplhr  mnl  St4iirH. 
Ihe  v;»iueiinl'<\/>^/i',  <$  '    ,  ami  A       ,  il>'|N'ndiiiK  <>h  the  proUtttt)*  of  the  nuisM  s 


78 


T  UK   O  II  U  I  T   OF    U  U  A  X  U  S. 


of  Jupiter  and  Sutiirn,  arc  f»ivon  on  pago  74  The  coniptitatioii  from  tlicso  data 
being  coiulucU-<l  in  tlu!  tamw  way  as  in  tlu>  c  asc  of  tli<>  terms  of  tlie  tirst  order,  it  is 
not  necessary  to  give  niueli  more  than  the  results.  These  are  sii  nvn  in  tlie  fol- 
lowing table.  The  indiees  to  the  left  represent  tliie  <-oefM(!ients  of  '.lie  mean  longi- 
tudes of  Uranus,  Saturn,  and  Jupiter,  all  counted  from  the  (n'rihelion  of  Uranus. 
Column  I'  gives  the  ratio  of  the  mean  motion  of  Uranub  to  the  coefficient  of  tho 
time  in  each  argument.  The  perturbations  of  the  comnu)n  logarithm  of  the  radius 
vector  arc  expresM-d  in  units  of  the  seventh  place  of  decimals. 


V 

lu 

0.4343  «i> 

U    S    J 

Hin 

VUH 

tl 

fi)» 

Hill 

0,      l,~l 

1, 

3. 
4, 

— 0.23fi4 
— 0.30i)f) 
—0.4480 
—0.8127 

+  0.002 
—0.020 
-f  0  004 
+  0.01(1 

0 

0 
— 0.0(1.1 
— 0.021 

0 
0 
0 

u 

0 

u 

0 
0 

—2,     2,-1 

o! 
1. 

0 

— 0.2!tr,0 
—0.4204 
—0.7  2.')  4 
— 2.(!420 
+  I.I10!I0 

—0.007 

— 0.  lo.s 
—  0  (111 

+0.1  r,4 
4-0  oo,> 

—.0  (Mil 
— .(I  (107 
—0,(112 
—0,2117 

-0.00,'. 

0 

—  2 

0 

1  1 

0 

0 
0 
0 

1  > 

u 

—3,     3,-1 

-2. 

—I. 

— 0.(!.5.')1 
—  I.81llt7 
+2.1115 
-1-0  ()7H« 

fO.Ol2 
1  0  17'> 
(-0,07s 
fo.007 

—  0,002 

— (i.dl.'l 
fO.:-.12 
)  It. 021 

0 

1  1 

—   2 
0 

0 
0 

t    :i 
0 

—J.     4,-1 

—2. 

—  1, 

-f0.7.'>» 

-f-(),«.10 
-1-0  301 

.—0.0.10 
f  0.013 

1  o.ool 

1  0.(I><1 

1  0.00.') 

— 0.003 

0 

—  1 

0 

f    1 
0 
0 

1,     3,-2 

•> 

4, 

—0.2170 
—0.2771 
— 0  :{s:),i 
—It  t.215 

_o,o()2 

—O.O.'.I 

—o.olo 
+  0.(t32 

—0.0.12 
1  (i,(i:;.'> 

(  0  O.'iO 

t  o.o.".2 

0 

-•  1 

0 
0 

—  1 

—  1 

-  1 

-  1 

0.     4,-2 

1, 

~0,3(;27 

— 0..5(i!i2 

—  1  3210 

1-4  ll.iO 

—0  010 

f  o.(i7."> 

— 0  :;i'.i 

— 0..M0 

—0  001 
— 0  I.M 
—  1  2'.i7 
— 0  l,>;t 

0 

—  :> 

—  ,1 

0 

'2 

1  II 
—  2 

— i! 

0, 

1. 

2, 

— 0.f)250 
— I.IO.M 
-)-10..'>ir.2 
-f0.!1132 
-1-0.4773 

--0  2ri3 

1.133 

_o,.'>ii; 
f  II  201! 
-f  0.012 

—0  (131 
—  o,(;17 
(Mi:(2 

— ;i  -j.ii 

—(I.I '.12 

—  3 

—  43 
♦    2 

—  2 
0 

0 

■i    7 
0 

—  :i(l 

—  2 

—4,     0,-2 
—3, 
—2, 
—  I. 
0, 

— 0.9»'t7 
—  lH.!l2rtO 

-)-l.o.''(ris 

4o..'>13(t 
-j-0.33!»3 

+  1.824 
-t  4O.C..'i0 
-(-t;,2:i7 
-(-o.4t.7 
+(».0(i7 

_.  0  ;i  1  (1 

-|O..Ml(l 
— 7.HI".(i 
_o  ."i:!!! 
—0.017 

+  10 

^32 

—(13 

0 

II 

1    *•• 

1  l» 

— 7'.l 

—  2 

0 

-4.     7,-9 

—2! 
—  1. 
—0, 

+OMM 
-|-0.3r)73 
-f  0  2i;32 
-f-0. 2(tH4 
+0.1724 

— tl  O.'iO 

— ii.oti; 

— 0  0I'> 
+0,000 

0 

— o.oo;i 

)  (t  (132 

1  0  (i:i2 

1  0  (107 

0 

f    1 
(     1 

♦     1 
0 
0 

0 

t  1 

■1  1 

0 
0 

^ 


TUK  OHBIT  OF  UKANUS. 


79 


ClIAPTKIl    V. 

COLLKCTION  AND  TRANSI'OKMATION  OF  TIIK  I'KECKDIXO  PJIRTURIJATIOXS 

OK  U  RAN  US. 


The  tcrniH  of  tlu.'  iicrtitrlmttonM  whirh  neither  contain  the  clnmrnts  of  tlin 
disturldn^  iiIaiictN,  nor  <li-|n'ii(l  on  the  scrular  variations  of  tlio  eccentricity  and 
perilicliou,  admit  of  being  greatly  ttiniplitied  Ity  u  slight  change  in  the  arbitrary 
elements.     These  terms  are  as  follows: 

(1)  In  the  longitude  of  Utanus 


Art  inn  of  Jii|iii(>r,  f  .11  -illtK 

Aflioii  of  Sutiirii,  -|-l«)  '.WMtl 

Ai-liuii  i>r  N<'(ituiic,  —  M.4Ui'i'ii 

T..I11I,  H ''■''<" 


4-j:>.(;ri7  sin  7  4- 1  .;i!)7  nin  2.7 

•  f    «..'i4.'>Hin  ^  -fit -Itll  hin  i!/ 

-f    OCnxin  ij  -fO.IMf,  din  iij 

-)-:i(.H'.)',)nin</  -\-\.'MiA\ii-2<j      — (i.(i82coH(/ 


1 ,  HM  con  g     —0. 0H7  rng  217 

4.7:J'>  O'lS  ij        — 0.  Id'.t  I  OK  'iij 


('2)  In  the  value  of  cos  i^p,  units  of  7th  place  of  decimals. 

A<-iii>n  or  Jiip'ti^r,         — liKlso  — I'.ii  con //  — :!:i  I'n.f  2;/  —  2Kiii(;  +  I  Kin  27 

Ariii>n  (>r  .Siitnrii,  —  :t.'i4:i  — H-l  cos  j;  — l.'>c<is2i/  — l.^niny 

Aciidu  of  X»'|»tiinc,        -f-     l'!S         —     Icosi/  —  I  ciiK  iiy  -I    iNlajr 

Tiiiiil,  — l:i('.l|  — C'TvDH'j  — 4',»i'o«2(/  — ICiHJuj  -f'liiin27 

l,et  ns  first  r(iii->id(>r  tiio  first  or  coiisfmit  term  in  the  perturlmtion  of  ea«h 
ro-onlinatc.  If  we  sM|i|tosc  a  change  of  hi  in  the  mean  nio'ion  of  a  |ilauet,  the 
eorrespiindiiig  change  in  i^p  will  Ix; 


ip: 


If,  tlieti,  we  iiicreusi-  the  mean  motion  of  Iranus  Ity  41".1."»1,  the  cnrrespcmdiiig 
rliange  in  Ap  will  he  iHOt.j,  imd  in  cos  >^ip,  —  IHO'Ji'i.  Subtracting  these  from 
th«'  above  jK'turb'itioiis,  the  serular  term  in  the  mean  motion  will  ilisaiipear,  and 
we  shall  hau-  for  the  c(in>r:tMt  tiTiu  of  cos  i^Ap 

j  l.-):U 

This  same  change  in  tlie  mean  motion  will  produce  a  sr^nlar  term  in  the  ('(pia- 
tion  of  the  centre  of  the  samf  nature  witli  that  prodnceti  by  the  s<'cular  variation 
of  the  perihelion.  The  differetices  of  the  values  of  the  secular  terms,  fouiul  b) 
!u'  two  inr^phrids  emploviil  in  ('li;ipters  H.  :md  TIT.  (trocerd-.  iVom  thr  fut  tliaf  in 
the  one  case  tin-  i  tb'it  of  tile  above  term  in  the  nnan  motion  is  included,  and  in 
the  other  excluded. 


£S 


80 


T  II  K   U  U  li  I  T   OF    U  U  A  N  U  S 


If  wo  Hiiliiluct  tlu!  rffcct  in  qiu-Rtion  when  iicrcsBiiry,  the  ri'iniiiii«lrr  will  In-  tlio 
cflft'ct  ut'  the  hcculur  variation  uC  tin*  longitude  uC  tlic  ixriliclion  of  I'ranus,  tu 
whii'h  we  Hliall  revert  presently. 

lA>t  us  next  introduce  Nueli  ii  ehnngp  in  the  ereentrieity  of  rninu)*  ns  Himll  |)ro- 
duee  the  terni  :)4".HiM)  Hin  «/,  and  uM-ertain  its  eti'vct  ou  the  other  turins,  l-'ur  tliiii 
purpose  we  luuitt  determine  te  by  tite  cunditiun 

3 


(2- 'c»)^c  =  34-.Hl)!) 


whii 


ch  gives 


;c=n".404  =  . 0000847. 
A  rlnmge  of  this  nmonnt  in  h  will  introduee  the  foUowin;^  t«'rnjs  in  h  nnd  ^p 

h  =  :14  ".H!)»  sin  (/  -f  2".()4M  itin  'i,j 
cos  ^■^p  =  *0  —  844  eon  <j  —  ftJ)  eos  'iij. 

Subtracting  these  terms  from  the  expressions  previously  found  we  have 

hvz=  —  0".  1 44  sin  •.>./  —  (;".(>M-.>  cos  fj  —  0".-J(i I  cos  Q;f. 
COS  4,^p  =i:  -(-  4 51 1  -f-  IGl  cos  ij  -\~  10  cos  '2i/  —  IG  sin  'j  -^  I  »in  2</. 

Again,  let  us  put 

c,^rt  =  3".a42^  .0000102, 
W(!  shull  have  the  elliptic  terms 

fiv  =  —  (;".f)82  cos  «7  —  0".:«)1  cos  'ij 
con  \J,^p  =  —  102  sin  ij  —  11  sin  2»/. 

Subtracting  these  ex])ression8  tlie  constant  terms,  independent  of  the  mean  hmgi- 
tude  of  the  disturbing  planets,  are  reduced  to 

hr  =  -  0".  144  sin  Qj  -f-  0".  1 30  cos  2</. 
cos <i,5p  =  4 ')  1 1  -f  1  (17  (OS (T/  -j-  10  cos  'iij  -\-  1 4fi  sin y  -|-  1 2  sin  'i>j. 
0.43429  ^p  =  IDG!)  -{-  IS  cosy  -f  4  cos  2</  +  G3  sin  r/  -f-  5  sin  2y. 

In  the  last  equation  we  have  introduced  the  constant  -|-.0000(M>rt  prtNluced  in  ^p 
by  the  combined  action  of  Venus,  tlie  Kartli,  and  Mars.  Tlie  effect  of  each  planet 
is  computed  by  the  approximate  formula 

^p=:  J  „,'(//;' +„/)aA7). 

Sinihir  ViiiladoiiD. 

The  following  inequalities  result  from  the  si-cular  variations  of  the  ecceiitririty 
nnd  longitude  of  perihelion  produced  by  each  of  the  disturbing  planets,  7' being 
the  time  expressed  in  centuries. 

From  the  variation  of  the  eccentricity 


ArUon  of  .lti))lt<-r, 
Ai'timi  of  Siiliirii, 
At'liiiti  uf  Neptune, 


ti). 


■  —I  'in;'/' Hill  7 

— ',1  Hi>7'hiii  ;/ 

— o.r)0'.'7'iiiii  <j 


— (Id"'.' 7' hill  '.''7 
— (l.Ti.'ls  7' hill  •-'</ 

— o.oaoy'hiii  'iij 


—  0. (Mi:.  7' Kill  .'I;; 

—II  ti:i:!/>iii  .lij 
— U  (iu-.!7'hiii  -.Ig 


*  * 


.. 


TUB  ORBIT  OF   UllANUH. 


81 


Action  nf  .lupltiT, 
Actiuii  of  Suiiirii, 
Actiuo  uf  N('|ttum', 


-[     li  T  fOK  (/ 


-I- 1  T  eon  8<; 
-f  Tif'cuii  ity 


The  soculur  vnriiition  of  thv  lon|{itiid(>  of  the  {icrihelion  is 


Artinn  of  .lii|iiti'r, 
Artioii  of  Suluni, 
Action  of  Ni'|ituno, 
Total, 


■\\t2.\r 

4-llH,»y 

+  r.i.ir 

««== +2111.(17' 


Thp  rffort  of  this  scnihir  varintinii  on  tlic  loiif^itudo  niul  radius  vertor  is 

It  II  II 

Action  of  JopiliT,        «('-— ll.4(!rcoM  j;  — (I.C.TI  T wh  2j  — 0.0J7  Tcob  3<; 

Anion  of  Siiliirii,                  —W.WTvmij  — ll.(!.'il  Tion  2</  — O.(i;{!t7'ro«  3«; 

Action  of  Ne|»tuiiP,               —  4.so7'ciiH»;  —  0  2«<|  7'coii  2</  —  O.ni«'i7'cob  :i</ 

Total,                         — 27.:t:7'coH  </  — I  C03  7'co«  2j/  — 0.  I02  7'co«  3jr 


Anion  rif  Jupiter, 
Action  of  Saturn, 
Action  of  Ncptuno, 


m. 


-I'-'iirHin  g 
-in  7'Hin  <j 
-  fiU  7'  Hin  g 


— Nrnin  Ig 
— f  7' Bin  2^ 
— 3  7'«in2j/ 


For  the  piirpoHO  of  convonicntly  t4(l)ii]atiii<;  the  prrttirliatinnH,  wr  tthull  pxprrss 
thrni  in  a  form  Hiniiliir  to  that  adoptrd  in  the  tlicory  of  Ncptinw*.  Let  uh  hchTt, 
from  tiio  tcnuH  of  tlic  |H>riotlir  pcrturhationH  produced  by  any  planet,  nil  thoM*  in 
wiiich  tlic  differcnco  iK'twcm  the  indices  •'  and  i"  is  tlie  same,  lor  exaniph',  in 
the  |H'rturbntiunK  of  tlie  loiij^itiide  produced  by  Jupiter,  let  us  consiiler  tlic  terms 


Ap=-}-l.V>(;i)sin(     -    ;) 

—  8.495  Bin  C^;—  0 
-f  l.lHisin(  g—'il) 
-f  O.ori  sin  (3y  - '2/) 

—  ().(»or)  sin  {'hj  —  ;}/) 
-f  ().onsin(4i/  — ;}/) 

These  terms  niny  be  expressed  in  the  form 


-f  0.002  cos  (      —    /) 

—  0.0!)'.>eos(27—    /) 
-fO..')l.'ieos(  ij~'il) 

—  0.00.')  cos  (%  —  '21) 

—  0.001  cos  (4«/  —  ;j/) 


>v=ih\\\ij  X 


-f  cos;/  X   ■ 


-1-0.094  sin    (v  — 0 
-|-  0.5',»0  sin  'i{ij  —  0 


—  'i.'i'.Hi  sin  (./  -  /) 
-4-  l.'.»')(isin%-  /) 
-f  0.(M»(i  sin  ;{(./  -  0 


—  4.7r»4eo»    ((/  —  /) 

—  l.lOHeos'%  — /) 
-f-0.01(icos%— /) 

—  0.090  cos    (,'/  —  /) 
•}   0,."»lOeos'%  — /) 


II        May.  1873. 


89  T  II  K   O  Kit  I T  0  F   U  II  A  N  U  H. 

In  gciicrul,  u  scrius  of  tornis  of  the  form 

Soi  lin  {iA  -f  «:/)  +  2  6,  cos  (I'A  -f  inj) 
-f  ia',  sill  {iA  —  ««/)  -f  S  t',  cos  («\4  —  «»/), 

may  be  put  in  tliu  form 

\^  (»»,  —  «',)  COS  iM  —  2i  (h,  —  li',)  Hill  iA  (  ttin  «(/ 
-f-  I-  ("(  +  "i)  "i"  '-1  +  i  (''(  4"  ''i) "'"  '■'^ { •'•"•"i/- 

All  the  fioriodic  terms  rontaiiiing  only  if  atul  /  in  the  nr^iimnits  miiy  he  put  into 
tluH  form  by  taking 

A=:t/-/, 

io  that  the  roeffioicntx  of  Hiii  /»■/  anil  vm  mj  may  all  ho  rxprcusnl  ns  n  function  of 
tilt*  HJn^lu  variable  ar^unHMit  A. 

Tlic  ]K-rtnrbation8  of  the  eh'mentfl  may  be  rnlnrcil  to  perturbations  of  the 
civortlinates  ex|»rc»sc<(l  as  the  sum  of  several  protiiirts  of  slowly  varying  functions 
into  the  sines  anil  cosines  of  the  multiples  of  <j.     W'v  have,  in  fact, 


iv  =  fl 


+  etc. 


+  etc. 


It  appears,  therefore,  that  all  the  perturbations  in  which  the  arguments  contain 
the  mean  longitudes  of  only  two  planets  may  be  ])ut  in  tin-  form 

iv  =  (tt.c.O)  -f  (r.r.  I )  cmq  -\-  (v.r.'i)  cos  2if  -|-  •'tc. 

-f  (r./i.I )  sin^  -f-  (r.M.'i)  sin  '2//  -j-  <'tc. 

J/,^p  =  ( p.r.O)  4-  ( ^}.l■.  \ )  COS  tj  -\-  ( p.c'i)  cos  'it/  -j-  etc. 

-{-  ( ('■"•  I )  sin  ij  -\-  { ^).l^,'i)  sin  'ij  -\-  etc. 

We  have  next  to  reduce  to  fbe  same  form  those  tenns  which  contain  the  mean 
longitudes  of  both  Jupiter  and  Saturn,  and  wbicli  are  given  on  |)age  78.  We  have 
here  twenty-four  terms,  each  greater  than  O'.OI.  As  most  of  these  terms  depend 
on  three  independ(-nt  arguments,  they  cannot  be  included  in  a  doidde  entry  table, 
while,  if  we  include  them  as  perturbations  of  the  hmgitude  in  tables  of  sin^jie 
entry,  we  shall  have  to  enter  twenty-two  tables  with  as  many  different  arguments. 
Hut,  by  taking,  for  the  ar^nmetit  .i,  the  miildle  one  in  each  series  of  argimients 
wliicb  depend  on  th(!  same  midtipl(>s  of  Jupiter  and  Saturn,  and  expressin;r  the 
terms  above  and  below  it  in  each  series  as  coefficients  of  sin  '/,  cos  ij,  sin  2y,  and 


■  o 


THE   ORBIT  OF  URANUS,  18 

COS  2j/,  wo  niuy  ridiioc  iho  iiuni»M«r  of  urKununtB  to  liglit,  niul  ihr  number  of  table* 
to  seventeen.     Consider,  for  inutunce,  the  terniH  of  the  second  series, 

-(U0Hsin(-i/4-'.».V-/)         -0.(Mncos(-j/-t-2.S'-J) 

—  0.0l4sin(  2S-J)         -().orico8(  '2S-J) 
4-().l()4Min(     ff-\-'iS-J)        —  O.'2(i7co8(     g-\-'2S-J). 

These  terms  may  be  ullowe«l  for  by  ndding  to  (r.o.O),  (i'.«.l),  (t'.c.I),  the  termi 

II  " 

(,,  ,..0)  =  —  0.014  sin  (2.V-  J)  -  0.0 1 2  cos  {'2S  —  J) 

(i'.«.l )  =  +  0.2<»0  sin  (2.V  —  J)-\-  O.'iTi  cos  CiS—J) 

(t'.f.l)  =  -f-  0.0:0  sin  {2S-J)  -  0.'i74  cos ('iS—J). 

From  the  perturbations  of  longitude  and  radius  vector  already  given,  we  readily 
find  the  following  values  of  (I'.c.O),  («'.«. I),  etc. 

Adii.n  0/  JiijilfiT. 

(»'.c.0)=-f53.0fi4  8in    A^  —0.004  cos    ^4, 

—  0.277  sin 'ivl,  -f  0.0;MJcos'.»/l, 

—  0.02r>8in;J^l, 

(».c.l)=-}-  2.226  sin    yl , —0.000  cos    Ay    ((•.«.1)=— 0.001  sin    yi, —4.7(54  cos    yl, 
_  1.2r)(}sin2y1,-f0.510eos2i1,  _0.&20sin2il,  — 1.108eos2.4, 

—  0.006  sin  :L1,  +0.016  cos  3.4, 

-11.467'  -I".  22  7' 

((•.c.2)=-|-  0.121  sin    i1,— O.OaScoH    y1,  (r.«.2)=— O-O-'jOsin    yl,— O.l75cos    yl, 

-f-  0.012  sin  2.4,  -0.01 4  cos  2.1,  +0.00H  sin  2.4,  +0.042  cos  2.4, 

4-  0.029  sin  ;L4,  —0.034  cos  3/1,  -f0.034  sin  3.4,  +0.035 cos  3.4, 

-0".6r/'  -0".07r 


(«.c.3)=-  0.04  7' 


(,..^.3)=^  _().()(),-,  7' 


(().c.O)=  +  1127cos    yl, 
+       4  cos  2.1, 

(p.f.l)=~       2 sin    .4, +.'.7  COM    .4, 
+     lO  sin  2yl,  —23  cos  2.1, 
+  137' 

(p.c.2)=+       7  cos    yl, +17' 


(().«.l)=  +  10Ssin    yl, +   2co8    yl, 
+   26  sin  2.1,  +12cos2yl, 
-120  7' 

(p.«.2)==+     7  sin    yl,  -87* 


IMAGE  EVALUATION 
TEST  TARGET  (MT-3) 


/. 


1.0 


I.I 


liillll    112.5 


|50 

Ir- 


.4^    IM 


i.': 
ii 


2.0 


1.8 


1.25      1.4 

1.6 

: 

<« 6"     - 

► 

Photographic 

Sciences 

Corporation 


23  WEST  MAIN  STREET 

WEBSTER,  N.>    14580 

(716)  872-V503 


^ 


t.<s> 


S'^.y 


Qa 

H 


A 


84 


TUE  ORBIT  OP  URANUS. 


(«.o.0)=»j 


(t).C.l)  = 


Action  of  Saturn, 

// 

It 

'+20"MsinJ, 
_  0.062V 

+8.580  cos    A^ 

—  4.110  sin  2 A, 

— 0.009  cos  2.4, 

—  0.824  sin  3^, 

—0.019  cos  3^, 

—  0.228  sin  4A, 

—0.008  cos  4yl, 

—  0.074  sin  bA^ 

—0.003  cos  5.4, 

—  0.025  sin  6 A, 

//                               " 

[—  1.032*/         '^ 

—  2.121  sin  2^, 

—  0.392  sin  3.4, 

—  0.109  sin  iA, 

—  0.038  sin  5^1, 

— U" 

e»s;:fj».^.(-..)-(i'«:e!«-"-+(+?:»y- 

—0.793  cos  2i4,                       —2.421  sin  2^,      +2.037  cos  2^, 
—0.308  cos  3^,                         —0.226  sin  3^,      +0.318  cos  3i4, 
—0.099  cos  4^,                       —0.055  sin  4^,      +0.097  cos  44, 
—0.027  cos  6i4,                       —0.027  sin  bA^      +0.038  cos  5J, 
.iir                                                      — 9".18r 

<"»)H:"::~)"".+t'';:Sr)™^-<""-''=(l''::Sy-".+(+'::"T)-^ 

—1.813  sin  2.4,  +0.631  cos  %A^ 
—0.177  sin  3^,  +0.013  cos  3^, 
—0.044  sin  4^,  +0.013  cos  4^4, 
—0.016  sin  bA^  +0.005  cos  54, 
— 0".65  2'  — 0".542' 


—  0.665  sin  2^4,  —1.847  cos  2.4, 

—  0.081  sin  34,  —0.153  cos  34, 

—  0.013  sin  44,  —0.044  cos  44, 

—  0.005  sin  54,  —0.016  cos  54, 


(t).c.3)  =  —  2.265  sin    4, 

—  1.163  sin  24, 
+  0.026  sin  34, 
+  0.005  sin  44, 
+  0.002  sin  54, 

— 0".04  2' 

—  0.126  sin    4, 

—  0.503  sin  24, 
+  0.053  sin  34, 


+5.656  ros  4, 
+2.956  cos  24, 
—0.063  cos  34, 
— 0.013  cos  44, 
—0.004  cos  54, 


(U.C.4)  = 


+0.329  cos  4, 
+0.378  cos  24, 
—0.032  cos  34. 


(p.r.O) 


=  +  106      sin    4, 
—      a      sin  24, 


(p.e.l): 


:  +  1364    sin    4, 
46    sin  24, 


sin  34, 
sin  44, 


+ 
+ 
+ 

+ 
+ 


(p.c.2)  =  - 


(p.c.3)  =  - 

+ 


+9sr 

279    sin    4,      — 
27    sin  24,      + 
3    sin  34, 

f7  2' 

61    sin    4,      — 
24    sin  24,      + 


761  cos    4, 

89  cos  24, 

19  cos  34, 

5  cos  44, 

43  cos    4, 

45  cos  24, 

9  cos  34, 

2  cos  44, 

103  cos    4, 
9  cos  24, 


24  cos    4, 
9  cos  24, 


(u.s.3)  = +5.656  sin    4,  +2.205  cos    4, 

+2.956  sin  24,  +1.163  cos  24, 

—0.063  sin  34,  —0.026  cos  34, 

—0.013  sin  44,  —0.005  cos  44, 

—0.004  sin  54,  —0.002  cos  54, 
— 0".03r 

(u.8.4)  =  +0.329  sin    4,  +0.126  cos    4, 

-10.378  sin  24,  .;-0.503  cos  24, 

—0.032  sin  34,  —0.053  cos  34, 


(p.s.l)  =  _ 


(p..s.2)  =  - 


(p.8.3)=- 


^ 

33  sin    4, 



1338  cos    4, 

+ 

31  sin  24, 

+ 

14  cos  24, 

+ 

7  sin  34,  . 

+ 

5  cos  34, 

+ 

2  sin  44, 

+ 

1  cos  44, 

—117  2* 

— 

103  sin    4, 

+ 

281  cos    4, 

+ 

9  sin  24, 

+ 

27  cos  24, 

+ 

3  cos  34, 

—8  2* 

._ 

24  sin    4, 

+ 

61  cos    4, 

+ 

9  sin  24, 

— 

24  cos  24, 

THE   ORBIT   OF  URANUS. 


86 


Action  of  Neptune. 
A,  =  g-r 

—0.08  cos  ^3 
—0.03  cos  2^3 
+0.23  cos  S.is 
4-0.06  cos  4^3 
—0.04  cos  5 At 
—0.01  cos  6^, 


,• 


(w.c.0)=— 39.66  sin  A^ 
—35.36  sin  2^,, 
+17.29  sin  Sylj 
+  3.91  sin  4yl3 
+  0.99  sin  5^3, 
+  0.42  sin  6^1^ 
+  0.19  sin  7^3 
+  0.09  sin  8.43 
+  0.02  sin  9.I3 
+11 

{v.cA)^ —  6.77  sin  A.^ — 0.53  cos   A3 

—  1.10  sin  2.I3 +0.07  cos  2-13 
+23.25  sin  3.43  +4.04  cos  3.43 
+  6.05  sin  4.43 +1.06  cos  4^3 

—  3.26  sin  5.43  — 0-66  cos  5^43 


O.lj  — 0.17  cos  6^ 
lA,  —0.05  cos  7.43 
8.43— 0.02  cos  8/I3 
9/I3  —0.01  cos  9yl3 


—  0.80  sin 

—  0.24  sin 

—  0.12  sin 

—  0.06  sin 

—  0.04  sin  lO/l, 

+1.99945e5i/ 
(tj.c.2)=— 0.43  sin  A^  —0.03  cos  A^ 
—0.03  sin  2.43 +0.01  cos  2.43 
+0.75  sin  3^3 +0.08  cos  3^3 
—0.10  sin  4^13— 0.08  cos  4.43 
—3.20  sin  5.43— 1.17  cos  5.13 
—0.83  sin  6.43—0.32  cos  6.4, 
+0.57  sin  7.43 +0.22  cos  7^ 
+0.14  sin  8.13 +0.06  cos  8.43 
+0.06  sin  9.43 +0-02  cos  9.43 
+0.03  sin  10.4,  —0.01  cos  10^3 
+0.11 722c  V 

A, 

3.43  +0.01  cos 

4.43  — 0.05  cos 

5,43  —0.02  cos 

6.43  —0.02  cos 

7.43 +0.25  cos    7  A, 

+0.11  sin    8.4, +0.07  cos   S.lj 

—0.08  sin   9^13-0.05  cos   9/I3 

—0.03  sin  10.43  —0.02  cos  IO.43 

+0.0071 4c  V 


((7.c.3)=— 0.02  sin 
+0.04  sin 
— 0.1 5  sin 
—0.08  sin 
—0.02  sin 
+0.46  sin 


3.43 

5.43 
6.43 


(y.s.l)=— 0.49  sin   ^3+  1.75  cos  A3 

+0.12  sin  2/I3—  2.48  cos  2^ 

+4.04  sin  3^3  —20.92  cos  3.43 

+1.07  sin  4^3—  5.42  cos  4^ 

—0.68  sin  5.43  +  3.47  cos  5^3 

—0.17  sin  6.43+  0.91  cos  6.4, 

—0.04  sin  7^3+  0.30  cos  7^, 

—0.02  sin  8^13  +  0.12  cos  8^3 

—0.01  sin  9.43  +  0.06  cos  9^ 

+  0.04  cos  10^3 

+1.99835Se 

(«7.8.2)=— 0.03  sin      ^3 +0.13  cos  A 

+0.02  sin    2.43— 0.16  cos  2^3 

+0.08  sin    3.43— 0.60  cos  3^3 

—0.08  sin   4^3 +0.15  cos  4J3 

—1.17  sin    5.43 +3.22  cos  5^13 

—0.32  sin    6^3 +0.86  cos  QA^ 

+0.22  sin    7^-0.57  cos  7yl3 

+0.06  sin    8.43— 0.14  cos  8^3 

+0.02  sin   9.43— 0.06  cos  9^, 

—0.01  sin  10.43  —0.03  cos  IO.43 

+0.117135e 

(,;.«.  3)=                           +0.02  cos  A^ 

+0.01  sin    3^1-0.04  cos  3^1, 

—0.05  sin    4.43 +0.15  cos  4^1, 

—0.02  sin    5.43  +0.08  cos  5.43 

—0.02  sin   6.43 +0.02  cos  6.43 

+0.25  sin   7/13—0.46  cos  7^3 

+0.07  sin    8.43— 0.11  cos  8^ 
—0.05  sin   9^+0.08  cos   9^, 
—0.02  sin  \^A.,  +0.03  cos  10^, 
+0.007 146e 


86 


TUB    ORBIT   OP   UUiNUS. 


Action  of  Neptune. — Continued. 
A,  =  g  —  1! 


(«>.c.4)=— 0.06  sin   9^  —0.05  cos   QA^    (/•.s.4)= 
—0.09  sin  10^3  —0.08  cos  10^, 
+0.00044e.'</ 


(p.c.O)= 


+  3  sin  3^3 


+227  cos  A^ 
+232  cos  2vl3 
—229  cos  3^ 

—  59  cos  4.4g 

—  17  cos  5^dj 

—  7  cos  6i4s 

—  3  cos  lAf 
+0.01018Je 


(p.c.l)=+  3  sin  A, 
+  2  sin  2.13 
+23  sin  3.43 
+  8  sin  4^3 

—  9  sin  5.43 

—  3  sin  6.I3 

—  1  sin  7^i, 


—  9  cos  IA3 
—141  cos  3^ 

—  39  cos  4^3 
+  43  cos  5  Jj 
+  13  cos  6^3 
+     5  cos  7^3 


— 0.43322  :e 


(p.c.2)= 


—  1  sin  3^ , 

—  4  sin  4.^3 

—  6  sin  5i43 

—  2  sin  6.^3 
+  3  sin  7^3 

— 0.03048:e 
(p.c.3)=— 0.000202.'e 


—  1  cos    A3 

—  6  cos  2 Jj 
+  6  cos  3^3 
+  5  cos  4.43 
+  17  cos  5^3 
+  7  cos  6^3 

—  7  cos  7/I3 


—0.05  sin  9^  +0.06  cos  9^, 
—0.08  sin  10^3  +0.09  cos  10^ 
+0.00044Se 


(p.«.l)=—  60  sin   .4' 

—  45  sin  2^3 
—107  sin  3Jg 

—  29  sin  4^ 
+  47  sin  5  J3 
+  15  sin  6  J3 
+     5  sin  7.^3 


—  3  cos   A3 

—  2cos2il3 
—23  cos  3^3 

—  8  cos  4.^3 
+  9  cos  5.^3 
+  3  cos  6^3 
+  1  cos  7.^3 


+0.43394eS^ 


(p.2)=- 


5  sin  A3 
8  sin  2vl3 
8  sin  3  J3 
5  sin  4^3 
17  sin  5^3 
7  sin  6^3 
—     7  sin  IA3 

+0.03053e3^ 
(p.s.3)=+0.00203e5«7 


+ 
+ 
+ 
+ 


+  1  cos  3^3 
+  4  cos  4.^3 
+  6  cos  5.4, 
+  2COS6J3 
—  3  cos  7  Jj 


Perturbations  of  the  Latitude. 

(The  secular  terms  being  omitted.) 

Action  of  Jupiter. 
II  II 

(&.c.0)=     0.024  sin    A^     —0.013  cos    A^ 

(fe.s.l)=— 0.420  sin    .^,     +0.494  cos    J, 
(i.c.l)=+0.494  sin    A^     +0.420  cos    A^ 

(6.s.2)= +0.004  sin  2.41     —0.017  cos  2Ay, 
(i.c.2)=— 0.017  sin  24i     —0.004  cos  2 J, 


t 


THE   ORBIT   OP  URANUS 


87 


Action  of  Saturn. 


\ 


(b.c.oy. 

=—0.08  sin    A3 

—0.03  cos   4 
—0.03  cos  24 

w 

II 

—0.01  cos  34 

(6.. 

.1)= 

=+1.34  sin    A 

+2.88  cos    Ai 

(i.c.l)= 

- — 1.56  sin    At 

+2.50  cos   4 

—0.01  sin  2Ja 

—0.10  cos  2.4j 

+0.02  sin  2At 

—0.10  cos  24 

+0.03  sin  3^ 

—0.06  cos  3^j 

—0.04  sin  3A 

—0.06  cos  34 

-1-0.02  sin  4^2 

—0.03  cos  4^j 

—0.02  sin  4^ 

—0.02  cos  44 

(b.s 

•2)  = 

=  —0.09  sin    Ai 

+0.10  cos    A^ 

{b.c.2)= 

=—0.09  sin    At 

— 0.05  cos    4 

—0.05  sin  2^ 

—0.09  cos  2At 
—0.01  cos  3^ 

+0.07  sin  2At 
4-0.01  sin  SAt 

+0.02  cos  24 

Action  0/ 

Neptune 
(6.c.0)= 

II 
=+0.01  sin    A3 
—0.01  sin  2A3 
—0.01  sin  34 
+0.01  sin  4^ 
+0.01  sin  54 

II 
—0.04  cos    4 
+0.00  cos  2^3 
4-0.04  cos  34 
+0.01  cos  44 

// 

// 

-1-0.01  sin  64 

{h.8 

.1)= 

=  +0.13  sin    A3 

+0.16  cos    A3 

(l.c.\)= 

= — 0.57  sin    A3 

+0.04  cos    4 

+0.09  sin  2A3 

+0.26  cos  2^3 

—0.39  sin  2^3 

-f0.06  cos  2^3 

-f  0.08  sin  3^ 

4-0.28  cos  3^3 

—0.33  sin  34 

+0.07  cos  34 

4-0.01  sin4^ 

4-0.03  cos  4il3 

—0.07  sin  44 

-j-0.01  cos  44 

—0.02  sin  5^3 

— 0.12cos5vl3 

+0.10  sin  54 

—0.03  cos  54 

—0.01  sin  6  A 

—0.03  cos  64 
—0.01  cos  74 

4-0.03  sin  64 
+0.01  sin  74 

(b.8 

.2)= 

=+0.01  sin    A3 

+0.05  cos   A3 

ib.c.2)= 

=—0.07  sin    A3 

—0.03  sin  3^3 

—0.09  cos  3^3 
—0.05  cos  44 
—0.04  cos  5^ 
+0.01  cos  IA3 

-J-0.09  sin  34 
+0.05  sin  44 
4-0.04  sin  5^3 
+0.01  sin  64 
—0.01  sin  74 

—0.03  cos  34 

■u 


Action  of  Jupiter  and  Saturn. 
(Terms  multiplied  by  the  product  of  their  masses.) 

N^=  2S—  J 

N.,=—  U-^SS—  J 
K3=—2U-\-iS—  J 
Nt=  SU-J-3S—2J 
N^=  2U-\-4S—2J 
JV„=  58— 'ir 

Nt^—UT-\-(SS—2J 
N^=—^U-\-lS-2J 


M  ( 


88 


THE   ORBIT   OP   URANUS. 


Action  of  Jupiter  and  Saturn  — Continued. 
(Terms  multiplied  by  the  product  of  their  masses.) 


(t).c.O)=+  0.08 
+  0.04 

—  0.01 

—  0.35 
f  —  0.55 
t  +40.65 

—  0.05 

(w.c.l)=+  0.06 
+  0.18 

—  0.03 

—  0.02 

—  0.44 
(  —  4.23 
\  +  8.06 

—  0.10 


sin  iVj 
siniV^ 
sin  JV, 

siniVj 

sin  JV, 
sin  Nj 
siniVj 

sin  iV, 
sin  Nj 
sin  J\r3 
sin  N^ 
sin  JVj 
siniVg 
sin  JV, 
sin  iVa 


+  0.51  cos  N^ 
4-  0.01  cos  JV3 
4-  0.05  cos  iVi 

—  1.30  cos  jv; 

—  0.03  cos  iVg) 
—10.50  cos  JVj  i 
+  0.03  cos  iVg 

—  0.21  cos  Ni     {V.8 
+  0.01  cos  iVj 

-f  0.08  cos  iV^3 
-I-  0.09  cos  iVi 

—  0.61  cos  iVj 

—  3.87  cos  ^;) 

—  8.38  cos  iV,) 
+  0.03  cos  iV, 


l)=+0.26  s 
—0.04  s 
+0.08  s 
—0.02  s 
+0.30  s 
f  +2.64  s 
t  +7.35  s 
—0.04  s 


njv; 

niVe 
niV, 


+0.27  cos  Nt 
—O.ncosNt 
+0.03  cos  N^ 
+0.08  cos  Nt 
—0.58  cos  N^ 
+4.64  cos  JV„ ) 
+4.41  cos  N,  i 
+0.00  cos  Nt 


(v.c.2)=  I  —0.24  sin  JV;  —0.22  cos  iV, ) 
t  +0.47  sin  JVj  —0.54  cos  iV,  ) 
(p.c.O)i=+l  1  sin  Ni    —3  cos  JVj 


(v.s,2)=  (  +0.16  sin  N^  +0.26  cos  iV„  1 
I  +0.54  sin  N,  +0.47  cos  N,  j 


Two  of  these  arguments,  namely,  5S — 2J,  and  —  Sg^GS — 2J,  are  of  very 
long  period,  that  of  the  first  being  about  880,  and  that  of  the  second  about  1590 
years.  It  will,  therefore,  be  convenient  to  tabulate  them  both  as  functions  of  the 
time  for  the  time  during  which  the  theory  is  to  be  used.  To  make  their  effect  as 
small  as  possible  during  the  period  for  which  the  provisional  ephemeris  is  to  be 
computed,  we  shall  suppose  the  longitude  of  epoch,  mean  motion,  and  longitude  of 
the  perigee  to  be  affected  with  the  negative  of  the  following  corrections : 

he=  +  27.27, 
^Tt  =  +  27.27, 
5n  =  —   0.1172. 

Reducing  these  corrections  to  corrections  of  the  co-ordinates,  and  adding  them  to 
the  terms  of  long  period  in  the  true  longitude  and  logarithm  of  radius  vector,  we 
shall  have  for  these  terms. 


(w.c.O)  =  —   0.546  sin  JV; 
+  40.650  sin  ^^ 


0.032  cos  A^„ 
10.500  cos  N,  +  27.27  —  11.72  T 


(«.s.l)  =      2.63  sin  JV„  +  4.64  cos  N^  +  7.35  sin  N,  +  4.42  cos  JV, 

(D.c.l)  =  —  4.22  sin  N,  —  3.87  cos  N^  +  8.06  sin  iV,  —  8.39  cos  N,  —  1.107* 


THE   ORBIT   OF   URANUd. 


89 


(».8.2)  =  +  0.16  sin  N^  +  0.26  cos  N,  +  0.54  sin  JV,  +  0.47  cos  N, 

(».c.2)  =  —  0.24  sin  N^  —  0.22  cos  N^  +  0.47  sin  N,  —  0.54  cos  N,  —  0".07  T 

(p.c.O)  =  4-   2  cos  iVe  +  10  sin  N.,  +  32  cos  N^  +  22 

(p.«.l)  =  —  41  sin  A';  —  43  cos  N^  +  82  sin  N,  —  85  cos  N,—  12T 

(p.c.  1)  =  —  29  sin  N^  —  45  cos  JV;  —  73  sin  iV,  —  44  cos  J\^, 

The  values  of  those  and  of  the  other  socuhir  terms  and  terms  of  long  period  for 
the  period  during  which  Uranus  has  been  observed,  are  given  in  the  following 
table : 


(ac.O) 

Nfpluiie 

.TnpitHr  and  Saturn 

Sum. 

(long  per.) 

(long  per.) 

1700 

ti 
+85.54 

// 
+4.86 

It 
+90.40 

17.50 

+38.45 

+1.90 

+40.35 

1760 

31.25 

1.48 

32.73 

1770 

24.80 

1.11 

25.91 

1780 

19.09 

0.80 

19.89 

1790 

14.12 

0.54 

14.66 

1800 

9.89 

+0.33 

10.23 

1810 

6.42 

0.17 

6.59 

1820 

3.69 

+0.07 

3.76 

1830 

1.71 

0.00 

1.71 

1840 

+  0,48 

—0.02 

+  0.46 

1850 

0.00 

0.00 

0.00 

1860 

+  0.27 

+0.06 

-f   0.33 

1870 

1.29 

+0.16 

1.45 

1880 

+  3.06 

+0..30 

+  3.36 

Values  op  i 

u.s.l) 

(1) 

(2) 

(3) 

(4)                  (r.) 

Jupiter 

Saturn 

Neptune 

Neptune        Jupiter  &  Saturn 

Sam. 

(aec.) 

(sec.) 

(sec.) 

(long  per.)           (long  per.) 

ff 

// 

ft 

// 

ft 

rf 

1700 

+  1.82 

+13.77 

+0.75 

—212.73 

—10.18 

—200.57 

1750 

+1.22 

+  9.18 

+0..50 

—141.97 

—  9.11 

—140.18 

1760 

1.10 

8.26 

0.45 

—127.76 

—  8.83 

—120.78 

1770 

0.97 

7.34 

0.40 

—113.. 55 

—  8.54 

—113.38 

1780 

0.85 

6.43 

0.35 

—  99.33 

—  8.24 

—  99.94 

1790 

0.73 

5.51 

0.30 

—  85.11 

—  7.93 

—  80.  .50 

1800 

0.61 

4.59 

0.25 

—  70.90 

—  7.61 

—  73.06 

1810 

0.49 

3.67 

0.20 

—  56.69 

—  7.28 

—  59.61 

1820 

0.36 

2.76 

0.15 

—  42.49 

—  6.94 

—  46.16 

1830 

0.24 

1.84 

0.10 

—  28.31 

—  6.60 

—  32.73 

1840 

+0.12 

+  0.92 

+0.05 

—  14.14 

—  6.26 

—  19.31 

1850 

0.00 

0.00 

0.00 

0.00 

—  5.91 

—     5.91 

1860 

—0.12 

—  0.93 

—0.05 

+   14.12 

—  5.57 

+     7.46 

1870 

—0.24 

_  1.84 

—0.10 

2S.23 

—  5.23 

20.82 

1880 

—0.36 

—  2.76 

—0.15 

_ 

+  42.32 

—  4.89 

+  34.16 

12      May,  1873. 


90 


THE   ORBIT   OP  URANUS. 


Valuks  of  (u.c.  1) 

.<'^ 

(2) 

(3) 

(•*) 

(5) 

Jupiter 

Saturn 

N«ptnne 

Nt-ptuna 

Jupiter  &rJAtarn 

Sam. 

(a«c.) 

(auu.) 

(»«o.) 

(long  p«r.) 

(long  per.) 

// 

// 

n 

/; 

II 

II 

1700 

+  17.19 

+10.07 

+7.20 

+37.63 

—1.98 

+76.71 

1750 

+  11.4(5 

+  11.11 

+4.80 

+28.07 

—2.28 

+53.70 

i7(;o 

10.31 

10.00 

4.32 

20.46 

—2.30 

48.73 

1770 

9.17 

8.89 

3.84 

24.10 

—2.45 

43.55 

1780 

8.03 

7.78 

3.36 

21.00 

—2.53 

38.23 

nno 

0.88 

0.07 

2.88 

18.05 

—2.01 

32.77 

1800 

5.73 

5.50 

2.40 

10.16 

—2.09 

27.16 

1810 

4.58 

4.44 

1.93 

13.23 

—2.76 

21.41 

1820 

3.44 

3.33 

1.44 

10.15 

—2.83 

15.53 

18;{0 

2.20 

2.23 

0.96 

6.92 

—2.90 

9.49 

1840 

+   1.15 

+   1.11 

+0.48 

+  3.53 

—2.97 

+  3.30 

1850 

0.00 

0.00 

0.00 

0.00 

—3.03 

—  3.03 

1800 

—  1.15 

—  1.11 

--0.48 

—  3.08 

—3.08 

—  9.50 

1870 

—  2.29 

—  2.23 

—0.96 

—  7.51 

—3.13 

—10.11 

1880 

—  3.44 

—  3.33 

—1.44 

—11.49 

—3.16 

—22.86 

(i>.».2) 

const.  = 

=  — 0".14. 

(1) 

(2) 

(3) 

(4) 

(5) 

Sum. 

// 

// 

II 

n 

// 

II 

1700 

+0.11 

+0.81 

+0.04 

—12.46 

—0.72 

—12.36 

1750 

+0.07 

+0.54 

+0.03 

— S.31 

—0.70 

—8.51 

1700 

0.06 

0.49 

0.03 

—7.48 

—0.69 

—7.73 

1770 

0.00 

0.43 

0.02 

—6.64 

—0.08 

—6.95 

1780 

0.05 

0.38 

0.03 

—5.81 

—0.66 

—6.10 

1790 

0.04 

0.32 

0.02 

—4.98 

—0.65 

—5.39 

1800 

0.04 

0.27 

0,02 

—4.15 

—0.04 

—4.60 

1813 

0.03 

0.22 

0.01 

—3.32 

—0.03 

—3.83 

1820 

0.02 

O.IC 

0.01 

—2.49 

—0.0 1 

—3.05 

1830 

0.01 

0.11 

+0.01 

—1.00 

—0.60 

—2.27 

1840 

+0.01 

+0.05 

0.00 

—0.83 

—0.58 

—1.49 

1850 

0.00 

0.00 

0.00 

0.00 

—0.57 

—0.71 

i8(;o 

—0.01 

—0.05 

0.00 

-1-0.82 

—0.56 

+0.06 

1870 

—0.01 

—0.11 

—0.01 

1.65 

—0.54 

0.84 

1880 

—0.02 

—0.16 

—0.01 

+2.48 

—0.53 

+  1.62 

(«.f.2) 

const.  = 

+  0".13. 

(1) 

(2) 

(3) 

(4) 

(■^') 

Snra. 

// 

n 

It 

It 

tf 

II 

1700 

+  1.00 

+o.ns 

+0.43 

+2.20 

—0.11 

+4.62 

1750 

+0.07 

+0.05 

+0.28 

+  1.08 

—0.13 

+  3.29 

17  CO 

+0.00 

0.59 

0.25 

1.55 

—0.12 

3.00 

1770 

+0.54 

0.52 

0.22 

1.41 

—0.12 

2.70 

1780 

+0.47 

0.40 

0.20 

1.26 

—0.13 

2.39 

1790 

+0.40 

0..39 

0.17 

1.11 

—0.13 

2.07 

1800 

+0.34 

0.33 

0.14 

0.94 

—0.13 

1.75 

1810 

+0.27 

0.26 

0.11 

0.77 

—0.14 

1.40 

1820 

+0.20 

0.20 

0.08 

0.59 

—0.14 

1.06 

1830 

+0.13 

0.13 

0.06 

0.39 

—0  14 

0.70 

1840 

+0.07 

+0.07 

40.03 

+0.20 

—0.15 

+0.35 

1850 

0.00 

0.00 

0.00 

0.00 

—0.15 

—0.02 

1800 

—0.07 

—0.07 

—0.03 

—0.21 

—0.15 

—0.40 

1870 

—0.13 

—0.13 

—0.00 

—0.43 

—0.16 

—0.78 

1880 

—0.20 

—0.20 

—0.08 

—0.08 

—0.16 

—1.19 

w> 


THE   Oil  BIT   OP   URANUS 


91 


(.•..S.3) 

(1) 

(2) 

(3) 

(4) 

(S) 

Jupiter 

Saturn 

Neptune 

Neptune        Jupiter  &  Saturn { 

Bum. 

(8W).) 

(8«C.) 

(««o.) 

(long  per.) 

(long  per.) 

// 

II 

// 

// 

// 

rr 

noo 

-f-0.01 

+0.05 

0.00 

—0.76 

0 

—0.70 

1750 

0.00 

+0.03 

0 

—0.51 

0 

—0.48 

17  GO 

0 

0.03 

0 

—0.40 

0 

—0.43 

1770 

0 

0.03 

0 

—0.40 

0 

— 0.3T 

1780 

0 

0.02 

0 

—0.35 

0 

—0.83 

1790 

0 

0.03 

0 

—0.30 

0 

—0.28 

1800 

0 

0.02 

0 

—0.25 

0 

—0.23 

1810 

0 

0.01 

0 

—0.20 

0 

—0.19 

1820 

0 

0.01 

0 

—0.15 

0 

—0.14 

1830 

0 

+0.01 

0 

—0.10 

0 

—0.09 

1840 

0 

0.00 

0 

—0.05 

0 

—0.05 

1850 

0 

0.00 

0 

0.00 

0 

0.00 

18G0 

0 

0.00 

0 

+0.05 

0 

+0.05 

1870 

0 

—0.01 

0 

+0.10 

0 

+0.09 

1880 

0 

—0.01 

0 

+0.15 

0 

+  0.14 

(«.p.3) 

(1) 

(2) 

O) 

(4) 

Sura. 

// 

// 

II 

// 

II 

1700 

-fO.OC 

+0.06 

+0.02 

+0.13 

+0.27 

1750 

4-0.04 

+0.04 

+0.02 

+0.10 

+0.20 

17C0 

0.04 

0.04 

0.01 

+  0.09 

+0.18 

1770 

0.03 

0.03 

0.01 

+0.08 

+0.i5 

1780 

0.03 

0.03 

0.01 

+0.07 

+0.14 

1790 

0.02 

0.02 

0.01 

+0.00 

+0.12 

1800 

0.02 

0.02 

0.01 

+0.00 

+0.11 

1810 

0.02 

0.02 

+0.01 

+0.05 

+0.09 

1820 

0.01 

0.01 

0.00 

+0.04 

+0.06 

1830 

+0.01 

+  0.01 

6.00 

+0.02 

+0.04 

1840 

0.00 

0.00 

0.00 

+0.01 

+0.02 

1850 

0.00 

0.00 

0.00 

0.00 

0.00 

ISfiO 

0.00 

0.00 

0.00 

—0.01 

—0.02 

1870 

—0.01 

—0.01 

0.00 

—0.02 

—0.04 

1880 

— O.Ol 

—0.01 

0.00 

—0.04 

—0.00 

For  the  uadi 

t;.S  VECTOR.       1 

t'^ALUES  OF  (|).( 

.0) 

0) 

(2) 

<3) 

(4) 

(5) 

Sum. 

II 

// 

II 

// 

// 

II 

1700 

+0.4 

+3.5 

—0.2 

+  104 

+  13 

+  181 

1750 

+0.3 

+2.3 

—0.1 

+110 

7 

+120 

1700 

0.3 

2.1 

0 

99 

0 

107 

1770 

0.2 

1.9 

0 

88 

5 

95 

1780 

0.2 

1.0 

0 

77 

4 

83 

1790 

0.2 

1.4 

0 

07 

3 

72 

1800 

0.2 

1.2 

0 

50 

2 

60 

1810 

O.I 

0.9 

0 

45 

2 

48 

1820 

0.1 

0.7 

0 

34 

+   1 

36 

1830 

+0.1 

0.5 

0 

22 

0 

23 

1840 

.0 

+0.2 

0 

+  11 

0 

+  11 

1850 

.0 

0.0 

0 

0 

—  1 

—     1 

18fi0 

.0 

—0.2 

0 

—  11 

—  2 

—  13 

1870 

—0.1 

—0.5 

0 

22 

—  2 

—  25 

1880 

—0.1 

—0.7 

0 

—  33 

—  3 

—  37 

«ll 


92 


THE   ORBIT   OF    URANUS. 


(p.«.l 

1                 count. 

-  +  63. 

(1) 

(2) 

(3) 

(4) 

(.'•) 

Jupiter 

Saturn 

Neptune    ' 

NnptuiiB 

Jupiter&i^atiirn 

Snm. 

(sec) 

(aeo.) 

(a«o.) 

(long  per.) 

(long  pur.) 

// 

// 

n 

« 

It 

/; 

1700 

+  180 

+  170 

+75 

+390 

—21 

+809 

1750 

+  120 

+  117 

+50 

+302 

—25 

+627 

1760 

108 

105 

45 

280 

20 

675 

1770 

96 

94 

40 

255 

28 

620 

1780 

84 

83 

35 

228 

29 

403 

1790 

72 

70 

30 

200 

30 

405 

1800 

CO 

59 

25 

170 

31 

346 

1810 

48 

47 

20 

139 

33 

285 

1820 

36 

35 

15 

106 

33 

222 

1830 

24 

23 

10 

73 

34 

158 

1840 

+  12 

+  12 

+  5 

+  37 

34 

95 

1850 

0 

0 

0 

0 

35 

+  28 

1860 

—  12 

—  12 

—  5 

—  39 

36 

—  41 

1870 

24 

23 

10 

80 

36 

—110 

1880 

—  36 

—  35 

—15 

—122 

—37 

-182 

(p.r.l) 

const.  — 

■  +  73. 

(1) 

(2) 

(3) 

(4) 

(5) 

Snm. 

// 

// 

// 

// 

// 

II 

1700 

—20 

—147 

—9 

+2241 

+104 

+2242 

1750 

—13 

—  98 

—6 

+  1490 

+  94 

+  1546 

1760 

—  12 

—  88 

—5 

1346 

91 

1405 

1770 

—10 

—  78 

—5 

1197 

88 

1265 

1780 

—  9 

—  69 

—4 

1047 

85 

1123 

1790 

—  8 

—  59 

4 

897 

82 

981 

1800 

—  6 

—  49 

—3 

747 

78 

840 

1810 

—  5 

—  39 

—2 

598 

74 

699 

1820 

—  4 

—  29 

—2 

448 

11 

557 

1830 

—  3 

—  20 

—1 

299 

68 

416 

1840 

—  1 

—  10 

—1 

+   149 

64 

274 

1850 

0 

0 

0 

0 

61 

+  134 

1860 

+   1 

+  10 

+  1 

—  149 

58 

—       6 

1870 

3 

20 

1 

298 

54 

—  147 

1880 

4 

29 

2 

—  447 

+  51 

—  288 

(p.s.2 

)            const. 

=  +5. 

(1) 

(2) 

(3) 

(4) 

tl 

(5) 

Sum. 

n 

// 

II 

;/ 

// 

1700 

+  12 

+  12 

+* 

+  28 

0 

+61 

1750 

+  « 

+  8 

+fi 

+21 

0 

+48 

1760 

7 

7 

5 

19 

0 

43 

1770 

6 

6 

5 

17 

0 

39 

1780 

6 

6 

4 

15 

0 

36 

1790 

5 

5 

4 

13 

0 

32 

1800 

4 

4 

3 

11 

0 

27 

1810 

3 

3 

2 

8 

0 

21 

1820 

2 

2 

2 

6 

0 

17 

1830 

2 

2 

1 

4 

0 

14 

1840 

+  1 

+  1 

+  1 

+  2 

0 

10 

1850 

0 

0 

0 

0 

0 

+  5 

1860 

_  1 

—  1 

—1 

—  2 

0 

0 

1870 

2 

—  2 

—1 

4 

0 

—  4 

1880 

—  2 

—  2 

—2 

-6 

0 

—  7 

"f 


* 


THE   ORBIT   OF   URANUS. 


93 


(p.c.2)            const. 

-+4 

(I) 

(2) 

W 

(4) 

(r>) 

JupitHr 

Saturn 

Nnptuiia 

Niiptuiie 

JupitHr  jiBatnrn 

Som. 

(seo. ) 

(8B0.) 

(»O0.) 

(long  per.) 

(long  per.) 

ti 

'/ 

It 

'/ 

// 

// 

noo 

—1 

—10 

0 

+157 

0 

+  150 

1750 

—1 

—  7 

0 

+  105 

0 

+  101 

17(10 

—  1 

_  G 

0 

!)5 

0 

!I2 

1770 

—1 

_  fl 

0 

84 

0 

81 

17H0 

—  1 

_  5 

0 

74 

0 

72 

1790 

—1 

—  4 

0 

03 

0 

02 

ISOO 

0 

_  3 

0 

52 

0 

63 

1810 

0 

_  3 

0 

42 

0 

43 

1820 

0 

—  2 

0 

31 

0 

33 

1830 

0 

—  1 

0 

21 

0 

24 

1840 

0 

_   1 

0 

+   10 

0 

13 

IS.'iO 

0 

0 

0 

0 

0 

+  i 

18(i0 

0 

+   1 

0 

—  10 

0 

—  5 

1870 

0 

1 

0 

21 

0 

_10 

1880 

0 

+  2 

0 

—  31 

0 

—25 

ReiJuced  E.ipres8iom  for  ihe  Latitude  of  Uranus. 

If  we  represent  by  F,,  T'i,  T^,  the  distances  of  Uranus  from  its  descending  nodes 
on  the  respective  orbits  of  Jupiter,  Saturn,  and  Uranus,  wc  find  tlie  following 
perturbations  of  the  latitude,  which  arc  independent  of  the  mean  longitude  of  the 
disturbing  planets. 

5/3=  — 0.0 11 4<  cos  F, 

—  0.0477<  cos  Fj 

—  0.0125/  cos  Fs 

-f  0.245 

4- 0.386  sin   «/  + 0.266  cos  g 

—  0.043  sin  2g  -j-  0.006  cos  2ij. 

To  find  how  far  the  last  five  terms  may  be  represented  by  simple  corrections  to 
the  elliptic  elements,  we  first  represent  the  cftect  of  minute  corrections  to  the 
inclination  and  node  of  Uranus  as  a  function  of  its  mean  anomaly.  Tutting  «  for 
the  argument  of  latitude  of  Uranus,  we  have  to  a  sufficient  degree  of  approxima- 
tion 

6/3  =  sin  m6<^  —  sin  <Ji  cos  iihO 
u  =<74-w+2esin(/ 

sin  M  =  —  e  sin  w 

-\-    cos  (J  sin   g  -\-    sin  o  cos    </ 
-\-  e  cos  (J  sin  Ig  -\-e  sin  o  cos  2g 


4 


94 


THE   OIiniT   OP   URANUS. 


COS  u  =  —  e  cos  (J 

-j-    cos  (J  cos   4) —    sin  (J  sin    g 
-\-  e  cos  (J  cos  '2(j  —  e  sin  o  sin  2(/, 

Substituting  these  values  of  sin  u  and  cos  u  in  the  expression  for  5/i,  and  putting 
sin  ^hO  =  6'0,  we  liavo 

5/3  =  c  cos  uh'O  —  e  sin  o  3<^ 

+  (  ''OS  u5(^  -(-    sin  u(VO)  sin    g  -\-  (  sin  u^^  —    cos  ufi'O)  cos    q 
-\-  (e  cos  o5^  -\-  e  sin  w6'0)  sin  2^  -f"  (»■  sin  (j(^<^  —  e  cos  urVO)  cos  2tf. 

To  represent  the  numerical  coefficients  of  sin  g  and  cos  if  in  Sjiif  wc  must  put 

cos  uh<p  +  sin  uh'O  =  ()".386 
sin  0)5^  —  cos  u6'0  =  0  .206, 

Since  u  ^  95°  3',  this  gives 


h'O 


0.231 ; 
0.409: 


5,iJ  =  — 0.013 

// 

+  0.380  sin    .7  +  0.200  cos   g 
-j-  0.018  sin  2,j  -j-  0.013  cos  '2g 

Subtracting  tins  expression  from  the  corresponding  terms  of  (\i,  wc  have  left 

^ii=^  0".258  —  0".()G1  sin  2g  —  0".007  cos  2y. 

The  first  term  of  this  expression  sliows  that  the  mean  orbit  of  Uranus  at  the 
present  time  is  a  small  circle  of  the  sphere  one-quarter  of  a  second  nortli  of  its 
parallel  great  circle. 
If  we  put 

V  =  longitude  of  Uranus  in  its  orbit,  referred  to  the  equinox  and  ecliptic  of 
1850,  we  have 

F,  =  «  —  127°  37 
V,  =  v  —  126  45 
Fa  =  w  —  155   32 

Substituting  these  values  in  the  first  three  terms  of  8;3,  and  multiplying  the  last 
term  by  the  factor  (1  +  /u)  by  which  the  adopted  mass  of  Neptune,  jj^^^,  must 
be  multiplied  to  obtain  the  true  mass,  we  find 

hiS  =  (4''.69  +  l".14,u)  Tcos  v  -  (5".24  +  0".52/i)7'8in  v. 

To  these  terms  must  be  added  those  which  arise  from  the  motion  of  the  ecliptic. 

In  the  absence  of  any  exhaustive  investigation  of  the  obliquity  and  motion  of 
the  ecliptic,  I  adopt  the  elements  of  Hansen,  employed  in  his  "  Tables  du  Soleil," 
because  they  are  a  mean  between  the  results  of  others,  and  are  very  accordant 
with  recent  observations.     The  secular  motion  of  the  obliquity  there  employed  is 

—  46".  78. 


*V 


iS 


THE   ORHIT   OF  URANUS. 


95 


IIiiiiHon  mentions  —  5".39  ns  the  corrospomUng  motion  at  the  equinox  of  1850, 
found  by  Olufson,  but  I  cunuot  reproduce  tliis  result  from  the  sccuhir  diminution 
witii  nny  musses  of  Mercury,  Venus,  and  Murs,  wliich  seem  to  me  probable.  The 
expressions  in  terms  of  the  masses  given  by  Lc  Verrier  ure  {Aumles  ile  VObserva- 
tvire  Impcriul  <le  Paris,  tome  ii,  p.  101), 

Secular  change    =  -  47.59  -  0.52i'  -  28.90*-'  -  0.83»''' 
Mot.  at  eipiinox  =  +  5.89  +  0.021^  +    7.57^'  +  0.73/'. 

In  this  expression  the  masses  of  Mercury,  Venus,  and  Mars  are  represented  by 


l+v         1  + 


and 


1-f 


„.„.  ' '! .«.  respectively.      The  influence   of  admissible 

3,000,000'  401,847'  "    '  2,080,337'  ^^  '"-"^  _     ^' 

changes  in  the  masses  of  the  other  plants  is  insensible. 

From  the  researches  of  l-e  Verrier  on  the  motions  of  the  four  inner  planets  T 
conclude  that  the  following  arc  about  the  most  probable  distribution  of  the  correc- 
tions of  the  masses  necessary  to  produce  the  motion  of  the  obliquity  given  by 
Hansen,  namely, 

V  =-     I 

v'  =  —  .018 

v'"=-    iV 

These  values  give  for  the  motion  at  the  equinox  of  1850 

-f-  5".  43 

Introducing  the  secular  variation  of  these  motions  avo  have,  for  the  change  in 
the  latitude  of  any  celestial  body  near  the  ecliptic,  arising  from  motion  of  the 
ecliptic, 

h(i  =  (5'.43  T-\-  0''.19  T-)  cos  v  +  (4G".78r—  0".0G  T-)  sin  v. 

Combining  this  with  the  change  arising  from  the  motion  of  the  orbit  of  Uranus, 
we  find 

hl3=\(lO".l2  +  l".Uu)T-\-0".l9T^cosv 

^  J(41".54  —  0".52,u)  r—  0".06  ^ |  sin  v. 

We  may  represent  these  expressions  in  the  usual  way  by  secular  variations  of 
the  inclination  and  node  of  Uranus.  But,  owing  to  the  small  inclination,  and 
consequent  rapid  motion  of  the  node,  it  will  be  necessary  to  include  the  coefficients 
of  the  second  power  of  the  time.  On  the  other  hand,  no  distinction  between  r 
and  6  is  necessary.  Putting  ^  for  the  inclination  of  the  orbit,  0  for  the  longitude 
of  the  node  referred  to  the  equinox  of  1850,  and 

2?  =  sin  <J»  sin  0, 
q  =  sin^  cosO; 


we  have 


sin  /3      =  —  pcQ9V-\-  q  sin  v 
cos  /35/3  =  —  hpco9v-\-  6q  sin  v. 


i 


96  THEORBITOPURANUS. 

From  the  expressions  for  p  and  q  wo  obtain 

cos  <^  Di(p  =  sin  0  D^p  -j-  cos  0  D,q; 
sin  ^  i><0  =  cos  0  Dfp  —  sin  0  Diq. 

And,  neglecting  (A<?> ^  X  sin  <f ,  we  have  farther, 

cos  ^E^tip  =  sin  ^  {Dfiy  +  sin  eD";p  +  cos  Oi)', j ; 

sin  ^D\0  =  —  2  cos^DfiB,^  -\-  cos  OZ/jj'  —  sin  OZ/V/- 

Since  <^  is  only  46'  wc  may  pnt  cos  ^  and  cos  (3  both  equal  to  unity  in  these 
expressions,  while  we  have,  for  1850, 

sinO=         0.9573 
cosO=         0.2890 
D,p  =  —  10".12  — r.i4« 
D,q   =+41.54— 0.52;^ 
.b^,p  =  _    0  .38 
L\q  =  —    0.12 
logsin<^=         8.129G06. 


The  above  formula;  then  cive 


D,^  =  +  2".3l     —  l".24,(t 
Dfi  =  —  31(i7".5  4-  12'.% 
D\^  =  +  0".26 
B'fi  =  +  5".6 


^  =  ^0  +  (2".3l 
0  =  0o  —  (31G7".p 


1".24//)  T'+O'.IS  7" 
12".fi'</)7'+  2.8  r. 


or,  adding  Struve's  precession,  we  have  when  0  is  counted  from  the  mean  equinox 
of  date, 

0  =  Oo  +  (1857".7  +  12".6//)  T-\-  3".9  T". 

Using  the  values  of  <^  and  0  given  by  these  expressions,  the  latitude,  secular 
variation  included,  will  be  given  by  the  expression 

sin/3  =  sin(f  sin  (t;  —  0). 

If  we  take  from  a.  table,  as  the  principal  term  of  the  latitude,  the  value  of  sin 
<jio  sin  (y  —  0),  the  secular  term  to  be  added  will  be 

S(2'.31  —  r.24«)  T-\-  0".13  r  j  sin  («  —  6). 

If  we  represent,  as  before,  by  «  the  variable  distance  of  the  perihelion  from  the 
node,  this  term  will  be  allowed  for  by  adding  to  (/>.s.l),  (/>.c.l),  etc.,  the  terms 


, 


THE   ORBIT   OF   U  11  AN  US. 


97 


Jk 


(b.c.0)  =  —  e  sin  u5<^, 
(h.K.l)  =  cosu^(J), 
(i.f.l)  =  sinut^^), 
(i.s.'i)  =  ecostaS<^, 
(6.C.2)  =       csinuf^^; 


Avlicrc 

Tutting  in  the  above  expressions 


l".->U)T-{-0''.ViT* 


,,,3=95^3' +  3459  "7; 
cos(j  =  — .0880  — .0167  r, 
siu(o  =  +  .99Gl  — .00157; 

we  find 

(!>.c.0)=:-(0".ll— 0".06/)7' 

{h..U)  =  —  (0  ••-'<>  -  0  -11")?'-  0".05r» 
(/>.c.l)  -  (-i  .30  -  1  .24-/) 7'+  0  .127'" 
{I>.k2)^  -0.01 7- 

(&.c.2)=       (0.11-0.06.7)7. 

We  have,  finallv,  to  consider  the  terms  of  long  period  in  hy;  nnd  <'/.;  which  luvvc 
been  omitted  froni  the  periodic  perturbations  produced  by  Neptune,  in  computing 
the  terms  of  <\i  on  page  61,  and  wliich  arc  as  follows: 

(^^1=       l".43cos(2/'—   J/)— 0''.39sin{2/'-    g) 

—  2  .12  cos  (4/'  —  2<j)  +  1 .00  sin  (4/'-  2*/) 
-f  0  .20  cos  (6r  —  3(7)  —  0  .04  sin  (G/'— 3i/) 

-f  constant  =  C.OO 

^•=      0".80cos(2?'—    J/)  — 2". 28  sin  (2/'-    j/) 

—  1  .06  cos  (4/'  —  2i/)  —  1  .85  sin  (4/'  —  2.'/) 
+  0  .04  cos  (6r  —  3.7)  +  0  .19  sin  (e/"— 3y) 

-\-  constant  =  0".364. 

For  the  period  during  which  Uranus  has  been  observed,  these  values  of  hri  and  tk 

may  be  replaced  by  tlie  following: 

,^>.  =  _0"807' 

^/•  =  + 0.277' 

which  arc  to  be  nudtiplied  by  the  factor  1  + ;,.     The  corresponding  perturbation 

of  the  latitude  will  be 

}^S  =  sin  v'v;  —  cos  V  k. 

Tutting  for  v  its  approximate  value 

V  =1  (/  -f-  (J  +  2t'  sin  g 
and  dtncloping  to  quantities  of  the  first  order  with  respect  to  the  eccentricities, 

we  have  .       ,      .      x 

sin  V  =  sin  {g  -f  ■  c)  +  e  sni  (2,7  + 1.>)  —  f  sm  u 
cos  V  =  cos  (j/ +  u) -i- c  cos  (2r/ +  (j)  —  c  cos  u. 

13       M»y.  1873. 


A 


98 


THE   0 11 B  1  T   O  1'    U  U  .\  N  U  S. 


Substituting  for  o  its  value,  1'2"45',  and  for  ly;  and  tJc  their  above  vahics  in  the 

expression  for  6ti,  we  find  that  the  terms  of  S3  in  question  will  add  the  following 

terms  to  (b.c.O),  (ft.s.l),  etc, 

(bj;,0)  =  —  .010  ^>7  +  .046  Ik  =  +  0".027'  (1  +  fi) 
(b.8.\)  =  +  .975  A>7  +  .221  hk  =  —  0  .727'  (1  +  n) 
(A.c.l)  =  4-  .221  hy,  —  .975  }k  =  —  0  .44 7^(1  +  ^) 
(/m.2)  =  +  .040  r^>7  +  .011  Ik  =  —  0  .047'  (1  +  //) 
(/>.f.2)  =  -  (b.c.O)  =  -  0  .027^  (1  +  fi) 

These  vahies  will  be  employed  in  the  construction  of  the  provisional  ephemeris, 

but  not  in  the  tables. 

Collecting  all  three  classes  of  terms  discussed  in  this    section,  we  have  the 

fallowing  constant  and  secular  terms  in  (/>.f.O),  [b.n.l),  etc. 

(ft.c.O)  =    +  0".26  +  (-0".09  +  O-OS:/)?" 
(h.8.1)  =  (—  0".92  —  0".Glr/)7'—  0".05  7^ 
(h.c.l)  =  (+  1  .86  —  1  .68,/)  7'+  0  .127"' 
(6.S.2)  =   —0.06  —  0.057' 
(6.C.2J  =   —  0  .01  +  (0  .09  —  .08 /)  T 

PosUloiia  of  Uranus  resulting  from  the  preceding  tlicory. 

The  next  step  in  order  is  the  preparation  of  an  ephemeris  of  the  planet  for 
comparison  with  observations.  As  this  provisi(mal  theory  is,  for  future  use,  super- 
seded by  the  tables  appended  to  the  present  work,  it  seems  unnecessary  to  enter 
very  fully  into  the  details  of  the  computation  of  the  ephemeris.  The  perturba- 
tions of  the  longitude,  logarithm  of  radius  vector,  and  latitude,  were  tirst  com- 
puted by  the  formulte  already  given. 

hv  =  (tJ.c.O)  -\-  {v.c.  1 )  cos  J/  -f  (!'.c.2)  cos  2//  +  etc. , 

4"  {v.8-V)  sin  g  -\-  (v.8.2)  sin  2g  -\-  etc., 

Jf5p  =  (p.c.O)  +  (p.c.l)  cos  .7 -f  (p.c.2)  cos  2.7 -j- etc., 

-f  (p.s.l)  sin  flf  +  (p-^.S)  sin  2^  +  etc., 

S.3  =  {b.c.O)  -\-  (J).c.l)  cos^  +  {b.s.l)  sin  g. 

Each  coefficient  (v.c.O),  {v.c.l),  etc.,  is  composed  at  most  of  the  following  quan- 
tities: 

1.  The  five  classes  of  secular,  long  period,  or  constant  terms,  the  separate  values 
of  wliich,  with  the  sum  of  all,  are  given  on  pages  89  to  93. 

2.  Periodic  terms  due  to  the  action  of  Jupiter,  Saturn,  and  Neptune,  given  on 
pages  83  to  87. 

3.  Terms  depending  on  the  product  of  the  masses  of  Jupiter  and  Saturn,  given 
on  page  88,  omitting  those  depending  on  iV„  and  N^,  because  they  arc  given  in 
column  5  of  the  terms  of  the  first  class. 

The  sum  of  the  perturbations  thus  computed  is  given  in  the  third  column  of  the 
following  ephemeris. 

An  approximate  value  of  the  perturbations  produced  by  Neptune  alone  is  inde- 
pendently computed  for  every  fourth  date,  and  tlie  result   is  given  in  the  fourth 


THE   ORBIT   OF   URANUS. 


99 


column.     The  secular  and  long  period  terms  are  liere  taken  from  columns  (3)  and 
(4)  of  the  tables  on  piigcs  89  to  93. 

The  elliptic  co-ordinates  were  then  derived  from  the  following  elements,  which 
are  a  little  different  from  those  employed  in  the  computation  of  the  perturbations. 


Elements  III. 

of  Uranus. 

7t,      168° 

15'    12".0 

e,         28 

25    29  .5 

0,        73 

11    58.0 

4,,          0 

46    20.0 

e. 

.0469436 

e,  (in  sec.)   9682".81 
n,  15426.196 

log«,  1.2828989 

Eed.  to  Ecliptic,  —  9".37  sin  2  (y  —  0) 

The  longitudes  thus  found  arc  corrected  for  lunar,  but  not  for  solar  nutation, 
and  the  results  are  given  in  the  fifth  column. 

The  column  "correction"  arises  in  this  way:  after  the  comparison  of  the  eplie- 
mcris  with  observations  was  nearly  completed,  it  was  found  that  some  errors  had 
crept  into  the  former,  the  most  importmit  of  which  was  the  employment  of  a  mean 
anomaly,  q,  corrected  for  secular  variation  of  the  perihelion  in  the  computation 
of  the  perturbations  from  the  preceding  formula;.  As  a  large  portion  of  the  com- 
putations on  the  provisional  ephemeris  had  been  made  by  assistants  furnished  by 
tlie  Smithsonian  Institution  and  Nautical  Almanac,  I  deemed  it  prudent  to  make 
a  careful  rccomputation  of  the  perturbations  for  every  si.xtli  date  during  the  entire 
period  of  the  modern  observations.  The  longitudes  actually  printed  m  the  fifth 
column  are  the  results  of  the  original  incorrect  computation,  while  the  numbers 
in  the  next  column  sliow  the  several  corrections  to  be  applied  to  obtain  the  results 
of  my  final  revised  computation. 

During  the  period  of  the  modern  observations  the  ephemeris  is  computed  for 
intervals'of  120  days,  and  the  selected  dates  are  all  exact  multiples  of  that  interval 
before  or  after  the  fundamental  epoch,  1850,  Jan.  0,  Greenwich  mean  noon. 
For  convenience  of  reference  the  dates  are  numbered  from  an  epoch  earlier  by  212 
intervals,  and  the  number  is  given  in  the  second  column. 

Between  1796  and  1801  no  observations  wortli  using  were  made  on  Uranus,  the 
ephemeris  has.  therefore,  not  been  extended  over  this  iiit(;rval. 


.L 


100 


THE   ORBIT    OP   URAxNUS. 


IleUOCENTKlC 

El'UEMEIUS  OF  UlLXNUS  FllO.M  THE  I'BEOKIH.MJ  TKOVIiSIONAl,  TlIEORY. 

[T 

lu  longitailea 

are  oorrwtfil  fur  lunar  but  not  for  solar  natatioD.] 

Dat». 
flrftfiiwioh 

No. 

Sum  of 

lieituiba- 

ApproxiniAlH 

ptTtiirbutioiiH 

producHd  by 

Neptune, 

Longltado. 

CorrfHition 

Latitude. 

Logarithm 
K.iclins 

lUtiUU  UUOll. 

tioiis. 

Tector. 

/        // 

// 

0      /       // 

// 

/       // 

lf.90,  Dec.    23 

4-4  28.5 

4-23C.1 

59  40  35.2 

—10    7.8 

1.2870828 

Dcf.    24 

+  4  28.5 

59  41   16.5 

—10    7.3 

1.2876789 

1712,  April    2 

+8  37.9 

155  24  42.0 

+45  58.7 

1.2026650 

April    3 

+8  .ST.O 

155  25  29.0 

+45  58.8 

1.2626640 

1715,  Mar.     4 

-ffi  5«.3 

4-136.2 

109  10  21.4 

+40    2.1 

1.2623890 

Mar.     5 

-J-r>  .58.2 

169  11     7.7 

+46    2.1 

1.2623893 

Mar.   10 

-f  (!  57.5 

169  15     0.0 

+46    1.7 

1.2623911 

April  2!) 

-)-(i  51.1 

4-134.8 

109  53  42.2 

+45  58.0 

1.2624006 

April  30 

f(!  51.1 

109  54  28.8 

+45  57.9 

1.2024102 

1748,  Oct.    21 

—0  28.9 

—  65.0 

310  13  19.3 

—41  27.0 

1.3001855 

Oct.     22 

—0  28.8 

316   13  58.8 

—41  27.9 

1.3001875 

1750,  Sept.  13 

+0     6.3 

—  51.3 

323  44  14.9 

—43  47.2 

1.3013714 

Sept.  14 

-fO     6.3 

323  44  54.0 

—43  47.3 

1.3013731 

Oct.     14 

fO     G.6 

—  50.8 

324     4  22.1 

—43  52.4 

1.3014105 

Oct.     15 

+0     6.6 

324     5     1.0 

43  52.5 

1.3014181 

Dec.      3 

-1-0     8.0 

—  49.8 

324  36  49.5 

—44    0.0 

1.3014850 

Doc.      4 

-1-0     8.0 

324  37  28.5 

—44    0.8 

1.3014872 

1753,  Doc.      3 

—0     5.6 

—  25.1) 

336  25     0.6 

—40     1.1 

1.3025789 

Di'c.      4 

— 0     5,6 

330  25  39.3 

—40    1.2 

1.3025790 

175f>,  Sept.  25 

—0  40.5 

—     0.4 

347  25  34.9 

—40    9.2 

1.3029303 

Sept.  2(i 

—0  49.5 

347  20   13.6 

—46    9.1 

1.3029363 

IICA,  Jan.    15 

-1-1   24.0 

16   13  35.0 

—38  36.9 

1.3003508 

Jan.     Ifi 

-1-1   24.0 

16   14  14.2 

—38  36.7 

1.30034HC 

17C8,  Dec.    27 

-f-1  33.1 

30     3  35.2 

—27  42.4 

1.2955331 

Dec.    2f) 

-fl   33.1 

+  100.9 

36     4  41.1 

—27  41.5 

1.2955274 

Dec.    31 

-f-1   33.1 

30     6     5.0 

—27  40.7 

1.2955216 

17fi9,  Jan.    15 

-1-1   33.5 

36  16     4.6 

—27  3,.2 

1.2954753 

Jan.    18 

+  1   33.5 

36   18     4,5 

—27  33.0 

1.2954659 

Jan.    21 

-fl  33.5 

+  106.9 

3(1  20     4,5 

—27  31.6 

1.2954566 

Jan.    24 

-f  1   33.5 

3(1  22     4,4 

—27  30.3 

1.2954476 

1781,  Jan.      1 

1 

+:i  57.46 

-1-178.9 

8(1  36   13.23 

+0.16 

+  11    3.03 

1.2785972 

May      1 

2  . 

4-3  53.41 

88     2  34.70 

+  12  11.14 

1.2780883 

Aug.  2it 

3  ' 

-f3  48.58 

89  29     7.41 

+  13  18.33 

1.2775834 

Dec.    27 

4  1 

4-3  43.09 

90  55  51.40 

+  14  25.14 

1.2770827 

1782,  April  2(J 

1 

4-3  37.18 

+180.4 

92  22  4(1.89 

-1-15  31.53 

1.2765870 

Anf;.  24 

'1  I 

43  30.97 

93  49  53.89 

+  10  .37.47 

1.2760968 

Dec.    22 

^  , 

-1-3  24.52 

95   17   12.51 

+0.18 

+  17  42.88 

1.2756124 

1783,  April  21 

8 

-1-3   18.03 

96  44  42.73 

+  18  47.73 

1.2751345 

Ang.   1!» 

!) 

-i-3   11.87 

4  180.1 

98   12  24.96 

+  19  51.93 

1.2746633 

Dec.    17 

10 

4-3     6.01 

99  40  19.00 

+  20  55.07 

1.2742012 

1784,  April  15 

11 

4-3     0.78 

101     8  25.13 

4-21  58.09 

1.2737412 

Aug.   I.i 

12 

42  56.24 

102  36  43.29 

+23    0.92 

1.2732907 

Doc.    11 

13 

4-2  52.82 

+178.6 

104     5   13,79 

+0.23 

-1-24    2.46 

1.2728469 

1785,  April  10 

14 

42  50.58 

105  33  56.61 

+25    3.14 

1.2724105 

Aug.     H 

15 

+  2  49.29 

107     2  51,36 

+20    3.05 

1.2719806 

Dec.      (i 

1(1 

-42  49., 35 

108  31  58,18 

+27    1.98 

1.2715564 

1780,  April    5 

17 

-42  50.53 

+  175.5 

110     I    16.88 

+  27  59.97 

1.2711405 

Aug.     3 

IH 

-f2  53,29 

111   30  47.68 

+28  57.02 

1.2707299 

Dec.       1 

in 

4-2  57.31 

113     0  30.11 

+  0.31 

-f  29  53.03 

1.2703265 

1787,   Mar.   31 

20 

43     2.38 

114  30  23.81 

+.30  47,92 

1.2699294 

July    2!) 

21 

+3     8.51 

+  171.0 

116     0  28.62 

+31  41,72 

1.2095394 

Nov.   2(1 

22 

4-3  15.48 

117  30  44.11 

+32  34.33 

1.2091,551 

\ 


+ 


T  H  K 

0  11  B  I  T 

OF    UK  AN  US. 

101 

IlKLlOCENTllIO  El'llE.MElllS  OF  UllANUS. — VunlillUed 

Datrt. 

Sam  of      lAPPf'f  im»l«  ( 

Logaritlim 

Ore«iiwicli 

i-.o. 

.     I         p«rturl)iitioii» 

Loiigitiiile.        Correction. 

Latitude. 

Radius 
vector. 

tui-aii  iiuoii. 

- 

tlOUB. 

Neptune. 

1 

'          "         i 

// 

o          /            // 

/        ft 

1788,  Mar.  25 

23  ' 

+3  23.02  ! 
3  31.30  i 

119     1     9.79 

-1-33  25.72 

1.2087775 

July    2:! 
Nov.   20 

24 

120  31   45.79 

34  15.81 

1.2084005 

25 

3  39.82  j 

-1-105.1 

122     2  31.13 

4-0.28 

35    4.58 

1.2080423 

1781),  Mar.  20 

2ri 

3  38.07  j 

123  33  25.90 

35  51.99 

1.2070852 

July    18 
Nov.    15 

27 

3  57.40  I 

125     4  29.49 

30  38.01 

1.2073353 

28 

4     5.97 

120  35  41.50 

37  22.50 

1.2009932 

1790,  Mar.    15 

29 

4   14.35 

-fl57.8 

128     7     1.70 

38    5.50 

1.2000592 

July    l;5 
Nov.    10 

30 

4  22.48 

129  38  29.88 

38  47.00 

1.2003330 

31 

4  30.00 

131    10     5.37       -1-0.25 

39  20.85 

1.2000107 

1791,  Mar.    10 

32 

4  30.95 

132  41   47.99 

40    5.04 

1.2057087 

July      8 
Nov.     5 

33 

4  42.95 

-1-119.1 

134   13  37.15 

40  41.58 

1.2054082 

34 

4  48.11 

135  45  32. (;2 

41  10.44 

1.2051215 

1792,  Mar.     4 

35 

4  52.33 

137    17   34.10 

41  49.53 

1.2(i48434 

July      2 

30 

4  50.03 

1,38  49  41.78 

42  20.89 

1.2045770 

Oct.    30 

37 

4  58.32 

-fl39.3 

140  21  54.47 

-1-0.40 

42  50.35 

1.2043210 

1793,   Fob.    27 

38 

4  59.50 

141  54  12.11 

43  18.00 

1.2040700 

Juno  27 

39 

4  59.85 

143  20  34.79 

43  43.81 

1.20,38448 

Oct.    25 

40 

4  58.84 

144  59     1.04 

44    7.75 

1.2030207 

1794,    F(!b.    22 

41 

4  50.80 

-fl2S.O 

140  31  32.77 

41  29.74 

1.2034220 

June   22 

42 

4  53.48 

148     4     7.47 

44  49.85 

1.2032327 

Oct.     20 

43 

4  49.13 

+  122.9 

149  30  45. S4 

-1-0.47 

45    7.99 

1.2030577 

1795,    I'el).    17 

44 

4  44.20 

151      9  27.73 

45  24.24 

Juno   17 

45 

4  3S.00 

152  42  12.55 

45  3S.44 

Oct.     15 
1801,  Jan.    17 

4(i 

4  31.04 

154   15     0.15 

45  50.04 

02 

+3  22.92 
3  20.59 

179     3  50.23 

-1-44  34.51 

1.2027599 

Mar.    17 

03 

+  57.3 

180  30  54.114 

44  12.H(i 

1.2028920 

Sept.  14 

04 

3  31.00 

182     9  57.40 

43  49.24 

1.2030302 

1802,  Jan.    12 

()5 

3  30.20 

183  42  57.23 

43  23.00 

1.2031922 

Mar.    12 

00 

3  41.88 

185   15  54.08 

42  50.34 

1.2033599 

Sopt.     9 

07 

3  47.78 

4-  43.7 

1.80  48  40.54 

-fO.35 

42  20.97 

1.2035387 

1803,  Jan.      7 

08 

3  54.01 

188  21   35.11 

•M  55.80 

1.203'' 281 

Mar.     7 

01) 

4     0.29 

189  64   18.55 

41  22.99 

1.2039280 

Sept.     4 
1804,  Jan.      2 

70 

4     0.43 

191  20  50.80 

40  48.28 

1.2041380 

71 

4    12.95 

-t-  30.7 

192  59  30.15 

40  11.79 

1.2043591 

Mar.      1 

72 

4   18.74 

194  31  50.83 

39  33.03 

1.2045891 

Aug.  21) 
Dec.    27 

73 

4  24.38 

190     4   17.13 

+  0.14 

38  53.72 

1.2048287 

74 

4  29.34 

197  30  30.24 

, 

38  12.21 

1.2050777 

1805,  April  20 
Aug.  24 
Deo.    22 

75 

4  33.'  ' 

-f   18.0 

199     8  35.97 

37  29.07 

1.2053300 

70 

4  37.^2 

200  40  33.91 

30  44.40 

1.2050037 

77 

4  40.11 

202   12  23.39 

1 

35  58.15 

1.2058808 

1800,  April  21 
Aug.   19 

78 

4  41.83 

203  44     4.25 

35  10.48 

1.2001074 

79 

4  42.. '.5 

-1-     5.9 

205   15  30.11 

'   -1-0.21 

34  21.31 

1.2004031 

Deo.    17 

80 

4  42.19 

200  40  58.07 

33  30.88 

1.2007089 

1807,  April  111 
Aug.  14 
Dec.    12 

81 

4  40.52 

208   IS   11.39 

32  39.05 

1.2070847 

82 

4  37.43 

209  49   13.98 

1 

31  45.90 

1.2074103 

83 

4  33.71 

—     5.0 

211   20     0.90 

30  51.03 

1.2077459 

1808,  April  10 
Aug.     8 
Deo.      (! 

84 

4  28.30 

212  50  49.02 

29  50.12 

1.2080922 

85 

4  21.03 

214  21  20.40 

+  0.25 

28  59.51 

1.2084491 

80 

4   14.32 

215  51   41.72 

1 

28     1.80 

1.2088100 

1801),  April    5 

87 

4     5.84 

—    15.4 

217  21  51.97 

27    3.03 

1.2091945 

Aug.     3 
Doc.      1 

8S 

3  50.03 

218  51   51.53 

1 

20    3.28 

1.2095829 

89 

3  47.07 

220  21   40.02 

25    2.00 

1.2099810 

1810,  Mar.   31 

90 

i      3  37.10 

221   51    19.10 

24     1.10 

1.2703905 

July    29 
Nov.   20 

91 

;      3  27.15 

—  24.3 

223  20  47.07       -|-0.23 

22  58.08 

1.270X094 

92 

+3  17.52 

224  50     4.83 

1 

4-21  55.38 

1.2712377 

102 


THE   ORBIT   OF   UUAXUS. 


UkUOCENTHIC  Fl'IIK.MElllS  OF 

UllANL'8. — 

Cunlinufd. 

Date. 
Greenwich 

No. 

Sum  of 
pfitiiibii- 

Ai>pioxiiiiatH 
pfrMirhalioiis 
pMMiuieil  by 

Loili^itUcle. 

Corieutloii. 

Liitltude. 

Lu}.'HI'ltlllll 

HailiiiH 

uuaii  iiuuu. 

tiuUK. 

veutiir. 

Nfptuiie. 

// 

/        ft 

// 

0 

1        II 

/        // 

1811, 

Mnr.  20 

93 

+3     7.99 

220 

19  12.04 

+20  51.52 

1.2716752 

July    24 

94 

2  59.12 

227 

48     9.00 

19  40.95 

1.2721215 

Nov.  21 

95 

2  50.91 

—32.1 

229 

10  55.87 

18  41.09 

1.2725700 

1812, 

Miir.   20 

9(i 

2  43.45 

2.30 

45  32.41 

17  35.80 

1.2730381 

July    18 

97 

2  37.20 

232 

13  59.13 

+0.11 

10  29.34 

1.2735072 

Nov.   15 

98 

2  31.98 

233 

42   15.55 

15  22.35 

1.2739838 

1813, 

Mar.   15 

99 

2  27.80 

—39.1 

235 

10  21.88 

14  14.92 

1.2744(i51 

July    13 

100 

2  25.05 

23(i 

38   18.20 

13    7.03 

1.2749507 

Nov.    10 

101 

2  22.98 

238 

0     3.95 

11  58.79 

1.2754409 

1811, 

Mar.   10 

102 

2  22.35 

239 

33  39.01 

10  50.21 

1.2759307 

July      8 

103 

2  22.54 

—44.0 

241 

1     4.03 

+0.22 

9  41.32 

1.2704300 

Nov.     5 

104 

2  24.24 

242 

28  19.59 

8  32.19 

1.2709372 

1815, 

Miir.     5 

105 

2  20.58 

243 

55  23.07 

7  22.80 

1.2774403 

July      3 

100 

2  30.00 

245 

22  17.17 

(1  13.39 

1.2779459 

Oct.    31 

107 

2  34.17 

—49.3 

240 

48  59.79 

5    3.80 

1.^784533 

181f., 

Feb.    28 

108 

2  39.00 

248 

15  31.47 

3  54.09 

1.2789(00 

Juno  27 

109 

2  44.51 

249 

41  52.08 

—0.01 

2  44.40 

1.27940S8 

Oct.    25 

110 

2  50.43 

251 

8     1.42 

1  34.82 

1.2799725 

1817, 

Feb.    22 

111 

2  50.75 

—54.4 

252 

33  59.49 

+  0  25.25 

1.2804709 

June   22 

112 

3     3.22 

253 

59   40.00 

—  0  44.19 

1.2809792 

Oct.    20 

113 

3  10.03 

255 

25  21.18 

1  53.44 

1.28147«8 

181S, 

Feb.    17 

114 

3   10.03 

250 

50  44. 4S 

3    2,48 

1.2819759 

Juno   17 

115 

3   22.90 

—54.7 

25  s 

15  55.79 

0.00 

4  11.23 

1.2824701 

Oct.     15 

110 

3  28.90 

259 

40  55.14 

5  19.73 

1.2S29599 

1810, 

Feb.    12 

117 

3  34.28 

2i:l 

5   4  2.25 

0  27.83 

1.283440  7 

Juno    12 

lis 

3  38.77 

202 

30   17.04 

7  35.52 

1.2839317 

Oct.      1(1 

119 

3  42.48 

—50.1 

203 

54  39.42 

8  42.77 

1.2844143 

1820, 

Feb.      7 

120 

3  45.03 

205 

18   49.19 

9  49.52 

1.284S927 

June     0 

121 

3  40.37 

200 

42  40.29 

—0.01 

10  55. 7(; 

1.2853070 

Oet.      4 

122 

3  40.38 

20  H 

0  .■5il.07 

12    1.40 

1.2X5.'*3!I1 

1821, 

Feb.      1 

123 

3  45.07 

—50.4 

209 

30     2.43 

13    fi.44 

1.2803087 

Juno     1 

124 

3  42.35 

270 

53  21.. 55 

14  10.88 

1.2807757 

Sept.  211 

125 

3  3S.08 

272 

10   27.90 

15  14.57 

1.2872402 

1822, 

Jiiii.    27 

1 20 

3  32.80 

273 

39  22.29 

10  17.58 

1.2,s77(i80 

May    27 

127 

3  20.18 

—50.2 

275 

2     4.25 

—0.10 

17  19.84 

1.2^81032 

Sept.  24 

128 

3   18.32 

270 

24  34.00 

18  21.32 

1.28,S(:214 

182;?, 

Jan.    22 

129 

3     9.08 

277 

40  52.38 

19  22.01 

1.2S90772 

May    22 

130 

3     0.24 

279 

8  59.05 

20  21.89 

1.2895311 

Sept.   HI 

131 

2  50.30 

—50.1 

2so 

30  54.82 

21  20.91 

1.2S99820 

1824, 

Jan.    17 

132 

2  40.10 

■>K1 

52  39.71 

22  19.10 

1.2904315 

May    Hi 

133 

2  29.74 

2S3 

14  14. k; 

—0.07 

23  10.40 

1.2908770 

Sept.   13 

134 

2  19.49 

2S4 

35   38.00 

24  12.78 

1.2913208 

182'), 

Jan.    11 

1.35 

2     9.25 

—55.5 

2S5 

50  52.92 

25    8.22 

1.2917000 

May    11 

130 

1   59.40 

2S7 

17  57.80 

20    2.71 

1.2921907 

Sept.     3 

137 

1   50.21 

2.SS 

38  53.43 

20  50.22 

1.2920290 

1820, 

Jan.      0 

138 

1   41.44 

2S9 

59  40.04 

27  48.71 

1.2930570 

May      0 

139 

1  33.58 

—54.5 

291 

20   1H.03 

—0.09 

28  40.20 

1.2934804 

Sept.     3 

140 

1   20.44 

292 

40  47.50 

29  30.70 

1.2938987 

1S2'7, 

Jan.      1 

141 

1    20.21 

294 

1      H,.^3 

30  20.10 

1.2943110 

May      1 

142 

1    14.89 

295 

21   22.15 

31*  8.00 

1.2947180 

Aug.  29 

143 

1    10.52 

—54.0 

29(1 

41   27.72 

31  55.92 

1.2951190 

I»ee.    27 

144 

1      7.22 

298 

1   25.72 

32  42,14 

1.2955130 

1828, 

April  25 

145 

1     4.79 

299 

21    10.45 

—0,07 

;!3  27.24 

1.2959002 

Au)r.  23 

140 

1     3.51 

300 

41      0.14 

34  11.25 

1.2902790 

Dee.    21 

147 

+  1     3.11 

—52.9 

302 

0  30.75 

—34  54.11 

1.29GG49C 

TIIK    OUIJIT   OF   UIIANUS. 


103 


m 


IIkLIoCENTIIIC  I-I'IIKMKUIS  01'  Utt 

ANUS.— 

Continued. 

Dftt«. 

Sii 

ni  of 

A|i|iroxiiiiat« 
perlartmtiotirt 
pniilucfil  l,y 

I/ocaritlim 

Orefiiwicli 
mtjnn  noun 

No. 

|iBrturba- 
tiuiiti. 

LoDgl 

uile. 

Correction. 

Latitude. 

Itailiud 
vector. 

Ni-piuiia. 

It 

// 

o        / 

" 

/; 

/        // 

182U, 

April 

20 

148 

+  1 

3.(18 

303  20 

fi.  73 

—35  35.77 

1.2970111 

Aug. 

18 

un 

5.15 

304  39 

30.08 

3(i  1(1.2(1 

1.2973030 

Dee. 

Hi 

150 

7.37 

305  58 

4(1. 9fi 

30  55.(10 

1.2977052 

1830, 

April 

15 

151 

10.14 

—52.3 

307   17 

57.34 

+0.08 

37  33.08 

1.2980308 

Aug. 

13 

1.52 

13.74 

308  37 

1.72 

38  10.50 

1.2983571 

Dee. 

11 

153 

17.(13 

309  55 

59.75 

38  40.13 

1.2980001 

^831, 

April  10 

154 

21.(19 

311    14 

51.(!8 

39  20.49 

1.2989038 

Aug. 

8 

155 

2(1.  or, 

—52.2 

312  33 

37.70 

39  53.54 

1.2992501 

Dee. 

(i 

15(1 

3(t.2(l 

313  52 

17.74 

40  25.29 

1.2995247 

1832, 

April 

4 

157 

34.13 

315   10 

51.(13 

+0.05 

40  55.74 

1.2997873 

Aug. 

2 

158 

37. 50 

31(1  29 

19.(17 

41  24.90 

1.3000379 

Nov. 

30 

159 

40.39 

—52.2 

317  47 

41.94 

41  52.05 

1.3002708 

1833, 

.M.ir. 

30 

KiO 

42.02 

319     5 

57.95 

42  19.10 

1.3005047 

July 

28 

101 

42.8(1 

320  24 

8.31 

42  44.17 

1.3007210 

Nov. 

25 

lfi2 

42.45 

321   42 

12.K9 

43     7.88 

1.3009280 

1834, 

•Mar. 

25 

1(13 

40.92 

—52.5 

323     0 

11.92 

+0.15 

43  30.24 

1.3011231 

.luly 

23 

1(14 

38.01 

324   18 

5.47 

43  51.18 

1.3013390 

Nov. 

20 

1(15 

34.01 

325  35 

53.99 

44  10.74 

1.3014851 

1835, 

Mar. 

20 

l(i() 

2s.7(i 

320  53 

37.59 

44  28.92 

1.3010515 

July 

IS 

1(17 

22.19 

—52.0 

328   11 

10.27 

44  45.09 

1.3018084 

Nov. 

15 

1(18 

14.89 

329  28 

50.92 

45     1.14 

1.3019507 

18311, 

Mar. 

14 

l(i9 

(1.50 

330  4(1 

21.41 

+0.19 

45  15.12 

1.3020!)00 

July 

12 

170 

57.31 

332     3 

48.20 

45  27.71 

1., 3022205 

Nov. 

!» 

171 

47.51 

—53.9 

333  21 

ll.7fi 

45  38.98 

1.302;!4S2 

1837, 

Mar. 

!» 

172 

.37.27 

334  38 

32.35 

45  48.84 

1.3024012 

July 

7 

173 

2(1.  (>9 

3i5  55 

50,40 

45  57.28 

1.3025050 

Nov. 

4 

174 

15.92 

337   13 

(1.1(1 

~ 

40    4.45 

1.. 3020022 

183.S, 

.Mar. 

4 

175 

+  « 

5.23 

—55. 1 

338  30 

20.17 

+0.17 

40  10.04 

1.3027503 

July 

•2 

17(1 

5.21 

339  47 

32.7(1 

40  14.3(1 

1.302S297 

Oet. 

30 

177 

15. 7  (i 

341     4 

43. si 

40  17.29 

1.3029008 

183'J, 

Feb. 

27 

178 

25. 5  S 

342  21 

54.35 

40  18.  S3 

1.3029034 

Juno 

27 

179 

34.82 

— 50.9 

243  39 

4.45 

4(1  19.00 

1.3030170 

Get. 

25 

180 

43.(18 

344  5(1 

14.08 

40  17.84 

1.3030030 

1840, 

Feb. 

22 

isl 

51.42 

34(1   13 

24.23 

+0.17 

40  15.28 

1.3030990 

June 

21 

182 

.. 

58.3(1 

347  30 

34.70 

40  11.29 

1.3031209 

0!t. 

lit 

1S3 

4.34 

—58.4 

348  47 

4(1.09 

40    O.Ol 

1.3031447 

1841, 

!•  el). 

1(> 

184 

9.47 

350     4 

58.23 

45  59.35 

1.3031528 

.luiie 

Hi 

185 

13.3(1 

351   22 

11.79 

45  51.:>3 

1.3031509 

Oet. 

14 

18(i 

1(1.11 

352  39 

2(1.97 

45  41.91 

1.3031392 

1842, 

Feb. 

11 

187 

17.73 

—(10.5 

353  5(1 

43.91 

+0.15 

45  31.07 

1.3031104 

Juuo 

11 

188 

IS.  i:t 

355   14 

2.89 

45  18.90 

1.3030817 

Oet. 

il 

189 

17.35 

25(1  31 

24.13 

4:    5.38 

1.  .3030351 

1843, 

Feb. 

(i 

190 

15.. 50 

.357  48 

47.74 

44  50.50 

1.3029707 

Juuo 

(i 

191 

12.83 

— (U1.7 

359    r, 

1 3.(15 

44  34.22 

1. 302900(1 

Oet. 

4 

192 

9.27 

0  23 

42.19 

44  10.04 

1..302822S 

1844, 

Feb. 

1 

193 

4.7(1 

1   41 

13.59 

+0.07 

43  57.59 

1.3027207 

May 

31 

194 

59.91 

2  58 

47.55 

43  37.23 

1.3020173 

Sept. 

28 

195 

54.(11 

—(14.9 

4  1(1 

24.34 

43  15.53 

1.3024947 

1845, 

Jail. 

2li 

19(1 

49.19 

5  34 

3.83 

42  52.43 

1.3023588 

May 

2(i 

197 

43.90 

(1  51 

4(1.0(1 

42  27.99 

1.. 3022095 

Sept. 

23 

198 

38.92 

a    9 

31.09 

42    2.22 

1.3020472 

1840, 

Jan. 

21 

199 

34.37 

—(17.7 

9  27 

19.04 

+0.15 

41  .35.11 

1.3018722 

May 

21 

200 

" 

30.55 

10  45 

9.82 

41     0.70 

1.3010851 

Sept. 

18 

201 

—0 

27.21 

1 2     3 

3.. 39 

—40  37.01 

1.3014801 

104 


THE    OllBIT  OP   URANUS. 


IIeliocentuiu  Ei'Iikmeuis  op  UiiANUS. — Continued.                                     1 

Date. 

Hum  of 

Approjciiiinte 

Logarithm 

Qre«iiwioli 
uieau  noun. 

No. 

p«itiirba- 
tiouH. 

Heitiirballons 

pi'odiicetl  by 

Neptniie. 

Longitiule. 

noireclloii. 

LutltuJu. 

UiKliUll 

vector. 

r        tf 

/; 

o                       // 

// 

/       (/ 

184T,  .Tan. 

10 

202 

— 0  25.04 

13  21     0.27 

—40    0.03 

1.3012758 

M.iy 

10 

203 

0  24.73 

—70.2 

14  39     0.21 

39  33.75 

1.3010544 

Si'pt. 

13 

204 

0  24.95 

15  57     3.01 

39     0.23 

1.3008219 

1848,  .Inn. 

11  ' 

205 

0  20.43 

17   15  10.40 

+  0.00 

38  25.40 

1.3005790 

May 

10  1 

200 

0  28.95 

18  33  21.30 

37  49.42 

1.3003203 

Sept. 

7  ' 

207 

0  32.75 

—72.8 

19  51   30.15 

37  12.18 

1.3000040 

1849,  .Ian. 

5 

208 

0  37.59 

21     9  65.34 

30  33.73 

1.2997923 

May 

5 

209 

0  43.47 

22  28   19.02   1 

35  54.10 

1.2995115 

Sept. 

2 

210 

0  50.27 

23  40  4S.13    1 

35  13.30 

1.2992224 

Dfc. 

31 

211 

0  57.74 

Y')  I> 

25     5  22.50    1 

—0.08 

34  31.47 

1.2989252 

1850,  April 

30 

212 

1     0.18 

20  24     2.38 

33  48.40 

1.2980203 

Aug. 

2S 

213 

1   15.11 

27  42  48.39 

33    4.37 

1.2983078 

Dec. 

2(i 

214 

1   24.38 

29     1  40.87 

32  19.00 

1.2979881 

1851,  April 

25 

215 

1  34.33 

—77.9 

30  20  39.78 

31  32.79 

1.2970015 

Aug. 

23 

210 

1   44.28 

31   39  45.91 

30  45.40 

1.2973287 

Dfc. 

21 

217 

1  54.21 

32  58  59.55 

—0.09 

29  57.09 

1.290989i 

1852,  April 
Aug. 

1!) 

21.S 

2     4.12 

34  18  20.89 

29    7.71 

1.2900430 

n 

219 

2  13.01 

—80.5 

35  37  50.43 

28  17.33 

1.2902910 

Dec. 

15 

220 

2  23.00 

30  57  28.19 

27  25.95 

1.2959338 

1853,  April  14 

221 

2  31.54 

38  17   15.00 

20  33.02 

1.2955701 

Aug. 

12 

222 

2  39.58 

39  37   10.77 

25  40.35 

1.2952015 

Dec. 

10 

223 

2  40.02 

—82.0 

40  57   10.17 

—0.07 

24  40.17 

1.2948270 

1854,  April 

9 

224 

2  52.04 

42   17  31.20 

23  51.08 

1.2944458 

Aug. 

7 

225 

2  57.77 

43  37  50.24 

22  55.08 

1.2940583 

Dec. 

5 

220 

3     1.00 

44  58  31.59 

21  58.25 

1.2930045 

1855,  April 

4 

227 

3     4.03 

—84.0 

40   19  17.49 

21    0.51 

1.2932045 

Aug. 

2 

228 

3     5.35 

47  40  13.93 

1     20    1.97 

1.2928582 

Nov. 

30 

229 

3     6.. 32 

49     I  21.20 

—0.11 

19    2.00 

1.2924449 

185G,  .Mar. 

29 

230 

3     3.95 

50  22  39.57 

8    2., 55 

1.2920239 

July 
Nov. 

2"      231 

3     1.24 

— Sli.l 

51   44     9.08 

17     1.07 

1.2915954 

24 

232 

2  57.02 

53     5  49.32 

10    0.03 

1.2911592 

1857,  Mar. 

24 

233 

2  52.95 

54  27  40.00 

14  57.72 

1.29071.55 

■July 
Nov. 

22 

234 

2  47.58 

55  49  42.98 

13  54.71 

1.2902043 

19     235 

2  41.05 

—87.9 

57   11  50.10 

—0.10 

12  51.08 

1.2898057 

1858,  Mar. 

19 

230 

2  35.18 

58  34  20.10 

11  40.89 

1.2893395 

July 
Xov. 

17 

237 

2  28.82 

59  50  54.59 

10  41.90 

1.2888059 

14 

238 

2  22.40 

01   19  39.71 

9  30.91 

1,2883853 

1859,  Mar. 

14 

239 

2  10.35 

—88.0 

02  42  35.14 

8  30.87 

1.2878981 

July 

Nov. 

12     240 

2  10.33 

04     5  41.38 

7  24.01 

1.2874049 

9  '  241 

2     5.17 

05  28  57.59 

—0.13 

0  17.90 

1.2809001 

18G0,  Mar. 

8     242 

2     0.07 

00  52  24.15 

5  10.84 

1.2804023 

July 
Nov. 

0 

243 

1   57.10 

— S9.1 

08   10     0.90 

4    3.42 

1.2858941 

3 

244 

1   54.37 

09  39  47.78 

2  55.74 

1.2853810 

1801,  Mar. 

3 

245 

1  52.01 

71     3  44.98 

1  47.83 

1.2848058 

July 
Oct. 

1 

240 

1   51.92 

72  27  52.38 

_  0  39.09 

1.2843408 

29 

247 

1  52.43 

—89.3 

73  52  10.31 

—0.19 

I-l-  0  28.00 

1.2838257 

1802,  Feb. 

20 

'  248 

1  53.51 

75   10  38.24 

1  37.00 

1.2833029 

June 

20 

1  249 

1  55.99 

70  41   10.05 

2  45.40 

1.2827788 

Oct. 

24 

;  250 

1  59.54 

78     0     5.40 

1 

3  53.95 

1.2822542 

1803,  Feb. 

21 

'  251 

2     3.98 

—88.5 

79  31     4.97 

5    2.40 

1.2817290 

June 

21 

i  252 

2     9.43 

80  50   15.07 

0  10.80 

1.2812047 

Oct. 

19 

,  253 

2  15.03 

82  21   30.01 

—0.04 

7  19.09 

1.2800819 

1804,  Fob. 

10 

1  254 

2  22.52 

83  47     7.95 

8  27.25 

1.2801009 

.luiie 

15 

255 

2  30.20 

—^7.2 

85   12  .'•)0.83 

9  .35.22 

1.2790421 

Oct. 

13 

250 

—2  38.39 

80  38   44.88 

+  8  42.95 

1.2791203 

TUK   OKU  IT   OF   UUANUS. 


105 


IIi-.iiiucKNTUic  Ki'iiKMKiiis  OF  UiiANi  1). — Voniltidfd. 

Datfl. 

Sum  of 

ApproxImRte 
pertarlmtinna 
prrdiiceil  by 

Nvptniie. 

Logarithm 

Oreeuwit'h 
muftn  iioou. 

No. 

perturl)*- 
tiona. 

Longitada. 

Correction.      Latitude. 

Radius 
vector. 

/       tt 

// 

0         /           // 

(/ 

/       If 

18G5,  Feb.    10 

2.'-)  7 

—2  47.06 

«8     4  00.26 

-f  11  52.20 

1.2786142 

Juno  lU 

2.')8 

2  55.93 

89  31     7.17 

12  57.54 

1.2781063 

Oct.      8 

2.')  9 

3     5.01 

—85.2 

90  57  35.61 

—0.09 

14    4.34 

1.2776029 

1866,  Feb.      5 

260 

3  13.91 

92  24  16.00 

15  10.74 

1.2771044 

Juno     5 

261 

3  22.61 

93  51     8.39 

16  16.70 

1.2766112 

Oct.      3 

262 

3  30.79 

95  18  13.06 

17  22.18 

1.2761237 

18CT,  Jnn.    31 

263 

3  38.4i» 

—82.5 

96  45  29.91 

18  27.15 

1.2756419 

Mny   31 

264 

3  45.25 

98  12  59.44 

19  31.54 

1.2751656 

Sept.  28 

2(;5 

3  50.96 

99  40  41.71 

—0.08 

20  35.37 

1.2746948 

1868,  Jim.    26 

266 

3  55.67 

101     8  36.54 

21  38.51 

1.2742299 

May    25 

267 

3  69.06 

_79.1 

102  36  44.31 

23  40.96 

1.2787709 

Sept.  22 

268 

4     1.02 

104     5     5.00 

23  42.75 

1.2733175 

1869,  Jnn.    20 

269 

4     1.71 

105  33  38.41 

24  43.69 

1.2728693 

Mny   20 

270 

4     0.81 

107     2  24.67 

25  43.81 

1.2724264 

Sept.  17 

271 

3  58.59 

—75.1 

108  31  23.. 55 

—0.11 

26  43.07 

1.2719888 

1810,  Jan.    15 

272 

3  54.96 

110     0  34.97 

27  41.44 

1.2715563 

Mny   15 

273 

3  50.23 

Ml  29  58.50 

28  38.77 

1.2711290 

Sept.  12 

274 

3  44.34 

112  59  33.91 

29  35.16 

1.2707070 

1871,  Jan.    10 

275 

3  37.90 

— C9.8 

114  29  20.79 

30  30.41 

1.2702903 

May    10 

276 

3  30.56 

115  69  19.11 

31  24.57 

1.2698791 

Sept.     7 

277 

3  22.61 

117  29  28.39 

—0.1 8 

32  17.57 

1.2694735 

1872,  Jan.      5 

278 

3  14.50 

lis  59  48.05 

33    9.31 

1.2690738 

May     4 

279 

3     6.19 

— C4.2 

120  30  17.95 

33  69.79 

1.2686S04 

Sept.     1 

280 

_2  57.05 

122     0  57.87 

+34  48.97 

1.2682934 

The  next  operation  would  bo  to  interpolate  these  co-ordinates  to  intervals  of 
time  suitable  for  the  computation  of  a  geocentric  ephemcris,  to  correct  the  longi- 
tudes for  solar  nutation,  and  then  to  compute  the  geocentric  right  ascension  and 
declination.  This  operation  has  not,  however,  been  completely  carried  out  except 
for  most  of  the  observations  before  ISIJO,  and  for  three  of  the  oppositions  observed 
since,  the  latter  being  computed  only  as  a  check  upon  the  accuracy  of  the  com- 
parisons. As  a  general  rule,  it  may  be  said  that  wherever  a  complete  geocentric 
ephemeris,  with  the  heliocentric  cphemeris  from  which  it  was  computed,  were 
available,  these  ephemcrides  were  made  use  of  in  a  manner  which  will  be  more 
fully  described  hereafter,  while,  in  all  other  cases,  the  geocentric  places  were 
computed  directly. 

It  may  also  be  stated  here  that  Hansen's  Tahka  tin  Solc.il  have  been  adopted  as 
giving  the  places  of  the  sun  to  be  used  in  computing  the  geocentric  places. 


H       May,  1873. 


100 


TUK   OIIHIT   OF   UIIANUS. 


CIIAPTER  VI. 

REDUCTION  OP  TIIK  OUSKUVATIOXS  OK  UUANUS,  AND  THEIR  COMPARISON 
WITH  THE  I'RECKDINU  THEORY. 

Tub  observations  of  Uranus  naturally  tlivide  thonisclvos  into  two  distinct 
classes.  (1)  The  purely  accidental  ones,  made  jjrevioiis  to  the  recognition  of  the 
planet  by  Ilerseliel  in  1781,  and  therefore  witliout  any  suspicion  on  tlu;  part  of  the 
observers  that  the  oltjeet  was  not  u  iixed  star,  and  ('2)  the  systematic  observations 
made  since. 

The  first  class  arc  nearly  all  so  uncertain  in  comparison  with  the  second  that  I 
have  hesitated  over  the  (piestion  of  employing  them  at  all.  If  nothing  but  a 
determination  of  the  elements  of  Uranus  were  called  for,  they  would  certainly  not 
be  worth  using,  since  these  elements  miy  be  determined  with  entire  certainty 
from  the  observations  which  have  been  nuide  during  the  entire  revoluticm  of  the 
planet  since  1781.  But  the  mass  of  Neptune  is  also  to  be  determined,  and  it  is 
at  least  possible  that  these  observations,  uncertain  though  they  are,  may  add 
materially  to  the  weight  of  this  determination.  I  have,  therefore,  determined  to 
include  them  all,  re-reducing  them  when  there  seemed  to  be  good  reason  so  to  do. 

The  earliest  observations  are  those  of  Flamstead,  ])Td)lishcd  in  the  Ilistoriic 
Coelestis.  The  observations  themselves,  as  printed,  together  with  the  principal 
elements  for  reduction,  are  given  in  the  following  tables. 

The  first  colunni  of  the  table  gives  the  name  of  the  star.  The  second  gives 
the  clock  time  of  transit  over  the  wire  of  the  quadrant  as  given  by  I'lamstead, 
The  time,  it  will  be  seen,  is  only  given  to  entire  S(>conds.  AVe  must,  therefore, 
exj)ect  to  find  a  probable  error,  of  which  the  mathematical  minimum  is  ()\2o, 
and  of  which  the  minimum  we  can  reasonably  expect  is  much  greater. 

Next  we  have  the  apparent  right  ascensions  of  the  stars  as  comjjuted.  For 
these  data  I  am  indebted  to  I'rof.  Coffin,  Superintendent  of  the  American 
Epiiemcris.  The  mean  ])laces  are  mostly  derived  from  the  "  Star  Tables  of  the 
American  Ephcmeris,"  and  from  the  two  Greenwich  Seven  Year  Catalogues,  while 
the  reduction  to  apparent  place  is  made  with  the  modern  constants. 

The  fourth  coliunn  gives  the  a])parent  clock  correcticm  for  sidereal  time,  in  which 
is  inchuled  the  eflect  of  deviation  of  the  instrument  from  the  meridian. 

The  clo(  k  keeping  mean  time,  the  errors  are  in  the  next  column  reduced  to 
those  of  sidereal  time  at  the  moment  of  the  transit  of  Urainis. 

The  next  two  columns  give  the  corrections  for  clock  rate,  and  for  deviation  of 
the  instrument  from  the  meridian,  as  inferred  from  the  observations  themselves, 
both  being  referred  to  the  time  and  position  of  the  transit  of  Uranus. 


'» 


t 


•  'W.mjHi.A.«R»U'-lMfflWaj- 


TIIK   ORBIT   OF   URANUS. 


107 


'» 


t 


In  the  last  column  wo  Imvo  the  seconds  of  conclnd.'d  com-ction  for  clcx 
instrument  to  be  upplied  to  tlie  observed  time  of  transit  of  Uranus. 


k  and 


IG'JO,  Ihwmber  23. 


Star. 

a  Arietis, 
7t  Arietis, 
a  Arietis, 
f  Arietis, 
5  Arietis, 
n  Tauri, 
Uranus, 
A  Tuuri, 


Tiino  of  Tp. 
Ii.    III.     n. 

7  48  40 

8  30  54 
8  ;j:}  17 
8  40  21 

8  52  44 
!)  27  47 

9  41  4!) 
y  45  3 


U.  A.  of  Htur. 

II.      III.         H. 


(.'lork 
h.    III.      B 


Iliijhl  AscciiHion. 

t". 
m.      «• 

58  29.3 


1  49  52.1   —5  58  47.9 

2  32  8.4  —5  58  45.6  58  33.9 
2  34  31.0  -5  58  4(5.0  58  34.7 
2  41   38.0  —5  58  43.0  58  33.0 

2  54     3.0  -5  58  41.0  58  33.0 

3  29  13.0  -5  58  33.4  58  31.1 


U. 

«. 
—1.1 
—0.7 
—0.7 
— O.G 
—0.5 
—0.1 


58  32.4 
5()  49.3 
58  39.0 


3  4(J  31.1  -5  58  31.9 
aViiginis,19  4  15  13  8  58.2  -5  55  1G.8 
o  Bootis,    19  58  31^  14     1  34.0  -6  50  57.5 

Hourly  rate  of  clock, 

Deviation  of  instrument  for  each  degree  of  Z.  D., 

Transit  of  Uranus, 

Correction  for  clock  and  instrument  (mean), 

Observed  11.  A.  of  the  planet. 


a  Arietis, 
7t  Arietis, 
a  Arietis, 
f  Arietis, 
h  Arietis, 
>7  Tauri, 
Uranus, 
A  Tauri, 


1690,  Dcccmhcr  23. 

Z.  D.  observed.     Refraction. 

2?)°  29'  10' 


35 
37 
31 
32 

28 
31 
30 


18  55 
41  0 
23  15 
55  55 
20  55 
52  35 
15  55 


+0'  33" 
4-0  41 

+0 
+0 

+0 


44 
35 
37 
31 
3() 
84 


16 
13 

20 
18 
23 


21  12  3 


Circle  reading  for  Uranus,  corrected  for  refraction, 

Equatorial  point  on  circle, 

Docliiiation  of  Uranus,  from  observation, 


Uranns, 
6  Virginia, 
e  Virginis, 
^  Virginis, 


1712,  Ajnil 

h.    m.      a. 

9  35  19 
12  0  19 
12  14  51 
12  32  11 


lil(jht  Ascension. 


li.      m.      B 

12  47  52.4 

13  2  31.5 
13  20     5.7 


Dev,     C 


B. 

-1.2 
+  1.7 
+2.9 
—0.2 
+0.5 
—1.3 


3i.6 
32.9 
32.5 
33.8 
33.0 
32.5 


0.0    —0.8     33.2 


— 0-.6 
—0.5 


h. 

9 

-5 
3 


m. 

41 


49.0 
58  32.8 
43  16.2 


DecUnafion. 

Dcfliiuition. 

21°  58'  53" 

9     4 

46  58 

4  37 

31  40 

6  53 


Eq.  point. 
51°  28'  36' 
28   40 

28 
28 
28 
28 


51 
51 
51 
51 
51 


42 
27 
12 

19 


51    28   32 

31°  53'  11" 

51    28    30 

+  19    35    19 


+47  33.4 
-1-47  40.5 
+47  54.7 


The  discordance  of  clock  errors,  and  the  time  whic^h  intervened  between  the 
transit  of  the  planet  and  tliat  of  the  first  star,  seem  to  render  an  accurate  reduction 
impossible. 


10b 


TIIK   Oil  BIT    OF   URANUS. 


nir),  MarrJi  4.     IHyht  Aurrnnion 
T. 

II.         III.         K. 

d  Lconis,         1 1  50  lU 
Urmius,  12  27     1 

6  Virgiiiis,      1'2  49  41         11  45  23.G 


II.  A. 

)l.         III.  K. 

10  45  52.5 


V. 
)i.    III.    *. 
1  4  2ti.5 


-1  4  17.4 

li.     m. 


0". 

20.5 

21.1 


Clock  time  of  transit  of  Uranus,  12  27     1 

Correction  for  clock  and  instrument,  — 1     4  20.8 
Ilight  ascension  of  Uranus  from  observation,  11  22  40.2 

1715,  March  4.     Vcctinalion. 


Z.  I).                        II.  Dec. 

d  Lconis,        40°  19'  40"  +1'  0"  +5°     8'    C 

Uranus,  46    33    10  +11 

iVirginis,      40    13   20  +1    0  5    14   17 

Circle  reading  for  Uranus, 

Equatorial  point. 

Observed  declination  of  Uranus, 


Eq.  point. 
51°  28'  46* 

51    28  37 

46°  34' 11" 
51  28  42 
+4  54  31 


1715,  Mirch  5.     Right  Ascension 

T. 

h.     in.      fl. 

rfl-eonis,         11  46  24 
Uranus,  12  22  59 

b  Virginis,      12  45  49 


II.  A. 

h.      m.       8. 

10  45  52.5- 


11  45  23.6 


0. 

h.    in.     R. 

— 1  0  31.5 


C 

25.5 


29.1 


-1  0  25.4 

h,  m.  8. 

12  22  59 

—1  0  27.7 

11  22  31.5 


Transit  of  Uranus, 

Correction  for  clock  and  instrument, 

Observed  right  ascension  of  Uranus, 

Tlie  large  apparent  clock  rate,  and  the  colons  after  the  time  of  transit,  both 
throw  doubt  on  this  observation. 

Declination. 
The  circle  readings  for  the  stars  are  the  same  as  on  the  day  preceding,  while 
that  for  Uranus  is  50"  less.     The  declination  is  therefore  50"  greater,  or 

+4°  55'  21". 

1715,  March  10.     lti(jhl  Ascension. 

C. 

h.    m. 

—0  40 
-0  40 


d  Leonis, 
p*  Leonis, 
Uranus, 
b  Virginis, 


T. 

h.      m.      a. 

11  25  58 

11  32  28 

12  1  42 
12  25  18 


R.  A. 

h.      m.      n. 

10  45  52.5 

10  52  25.1 

11  45  23.6 


2.9 


59.5 
58.1 

58.2 


Clock  time  of  transit  of  Uranus, 
Correction  for  clock  and  instrument. 
Observed  right  ascension  of  Uranus, 


-0  39  54.4 

h. 

12 

—0  39  58.6 
11  21  43.4 


m.    B. 

1  42 


TIIK  OIiniT  OF  URANUS. 


109 


UvcUiialio)!. 


v] 


tl  Loon  is, 
^i'  liconis, 
Uranus, 
b  Virginis, 


Z.  n. 
4(5°   ID'  JJ/J" 
47    M   M 
40    27     0 
4G    13   2Q 


11. 

+  1'  1" 

4-1  ;) 

+1  1 
-i-i  0 


Dci'. 
5°  H'     ()" 

6  14    17 


E.|.  rt. 
51°  'i<  -i'r 

51    21'  0!i 


Circle  reading  for  Uranus, 
Eciuatorial  point  on  circle. 
Observed  declinutiou  of  Uranus, 


51 

46 
51 

+5 


a  Lconis, 
Uranus, 
V  Virginis, 
17  Virginis, 
X  Virginis, 


1715.  Ajn-il  29 

T. 

ii.  III.  *. 
8  42  11 

8  50  44 

9  G  55 
9  43  38: 

11  32  48 


RitfJit  Ascension. 
n.  A. 


h. 
11 


ni.     •• 

G  28.3 


11  31   14.1 

12  8     G.5 

13  57  47.G 


C. 

Ii.    ni.      •. 

+2  24  17.3 

4-2  24  19.1 
^-2  24  28.5 
+2  24  59.G 


28  42 

2ir  1 

28  49 

U  48 


C 


8.7 

[6.4 
19.8 
33.1 


Ti.e  discordance  of  the  clock  corrections  mak.-s  a  satisfactory  determination  ot 
the  right  ascension  very  difficult.  1  deem  it  best  to  reject  the  doubt.l  o  bserva- 
tiou  of  17  Virginis,  and  the  discordant  one  of  .  Virguus.  'ihc  resuU  wdl  then 
be 


Observed  transit  of  Uranus, 
'^Jorrcction  for  clock  and  instrument, 
)bscrved  right  ascension  of  Uranus, 


h.    in.      «. 

8  50  44 
2  24  17,6 
11  15     1.6 


Z.  D. 

a  Lconis,       43°  52'  40" 
Uranus,  45    45    30 

v  Virginis,    43    20   20 
17  Virginis,  44    34    10 
X  Virginis,     GO    23     5 
Circle  reading  for  Uranus, 
Mean  equatorial  point. 
Observed  declination  of  Uranus, 


Declination 

B. 

-f  0'  55" 
4-0  59 
-fO  54 
-fO  5G 
4-1  41 


Dpc. 
7°  34'  51" 


8 
6 

—8 


7  11 
53  33 
55  42 


Ef .  Pt. 
51°  28'  26" 

51  28  25 
51  28  39 
51  28  G4 

45=  46'  29" 
b\    28  38 
4-3  42  9 


The  next  observations  in  the  order  of  time  are  two  by  Bradley,  discovered  by 

Mr.  Hugh  Breen,  but  still  unpublished.     The  following  are  tbe  results  as  given 

by  Mr.  Breer  in  the  AstronomiscJie  NachricJikn,  No.  1463. 

Mean  Time.  ^-  ^■ 

1748,  October  21,  7  6  18.4        21     4  37.93 

1750,  September  13,      10  8  57.8         21  40     0.23 


N.  P.  D. 

107  29 
104  42  33.9 


I 


no 


TIIK   ORRIT   OF    URANUS. 


Mr.  Brccn  remarks:  "The  right  ascensions  are  very  acciirute.  It  has  been 
assumed  tiiat  the  N.  P.  D.,  on  l"i5(),  September  13,  is  identical  with  //  ('apricorni, 
with  which  it  was  compared.  Tlie  first  observation  was  by  the  transit  instrument, 
and  the  second  by  tlic  quadrant." 

No  ground  is  given  for  the  above  assumption  respecting  tile  N.  P.  D.  for  tlie 
second  observation ;  it  may,  therefore,  be  omitted  as  valueless. 

In  the  year  1750  we  have  also  two  observatidns  by  Le  Monnier  at  Paris.  For 
these,  and  all  the  other  observations  by  the  same  observer,  1  shall  adopt  the  results 
given  by  Bouvard  in  the  C'oimaissince  dia  Temps,  for  1821,  p.  341,  with  tlie  cor- 
rections indicated  by  Le  Vcrrier,  in  Connahsance  iJcs  Temps,  for  1849,  pp.  125  and 
126.  The  necessary  uncertainty  of  the  observations  is  sucli  that,  consid:  ring  that 
Bouvard  reduced  them  with  the  star  positions  of  the  "  Fundumeula,"  scarcely 
anything  will  be  gained  by  a  new  reduction. 

1753,  December  3,  we  have  another  observation  of  right  ascension  by  Bradley. 
I  adopt  tlie  result  kindly  communicated  by  my  distinguished  friend.  Dr.  Auwers. 


1753,  December  3, 


Ii.    m. 

5  33. 


R.  A.  =  22  23  21.59 


1756,  September  25.     Observation  by  Mayer,  at  Gottiugen.     I  adopt  the  result 
given  by  Bessel,  in  Fundamenta  Astronomia:,  p.  284. 


li.     m. 

1756,  September  25,         10  12. 


R.  A.  =  348  0  54.5 
Dec.   =  —6   1  49.4 


The  following  is  a  tabular  summary  of  the  preceding  results,  with  their  com- 
parison with  the  provisional  theory.  In  the  computation  of  the  geocentric  place 
the  places  of  the  sun  were  derived  from  Hansen's  Tables.  I  am  indebted  to  Pro- 
fessor Coffin  for  a  duplicate  computation  of  the  geocentric  places  from  the  pro- 
visional ephemeris,  which  was  executed  by  Mr.  Joseph  A.  Rogers. 


-  ate. 

Right  Asceuslon. 

Deulinatiou. 

Correction  to  theory.                        1 

Obserratlon. 

i- 

Observation. 

£• 

R.  A. 

Deo. 

liOng. 

dl 

'ei 

J 

s 

r)-A. 

■  ep 

H 

II 

h   m     s 

s 

0           1           II 

II 

It 

II 

IfiflO,  Dec.  23 

3  43  lfi.2 

14.7 

+  19  35  19 

7 

+22 

+  12 

+24 

1.04 

+.027 

171.'),  Mar.    4  11  22  40.2 

38.7 

4  54  31 

48 

+22 

—17 

+28 

1.00 

0 

Mar.    5  11   22  31.5 

29.1 

4  55  21 

49 

+30 

—28 

+  4* 

1.00 

0 

Mar.  10  11  21  43.4 

41.0 

5     0  4i; 

5(! 

+30 

_  8 

+30 

1.00 

0 

Apr.  29  11  15     l.fi 

1.5 

-f   5  42    9 

U 

+  1 

—  2 

+  2 

1.04 

+.0.30 

1748,  Oct.  21  21     4  37.03 

35.42 

+37.0 

+39.3 

1.015 

+  .0.50 

1750,  Sept.  13  21   40     0.23 

o          /         " 

57.90 
II 



+35.0 

+35.7 

1.05 

+  .022 

Oct.  14  324   15  24. f) 

:47.r, 

—15     1  40.4 

47.0 

+37.0 

+  0.0 

+35.9 

1.03 

+  .045 

THE   OUBIT   OF   URANUS. 


Ill 


1 


Date. 

Right  Asoenalon. 

Declination. 

Correction  to  theory. 

•     1 
ObserTatlon.        o     , 

Observation. 

b 

R.A. 

Dec. 

Long. 

01 

m 

69 

^  ! 

/ 

1         " 

;  o      '      '1   1    // 

O        1          It 

tf 

ti 

ItoO,  Doc.    3  324  34  53.5  15.4 

_14  53  20.2 

32.4 

438.1 

412.2 

438.8 

0.98 

+.047 

h    111        8      1      s 

1753,  DfC.    3  22  23  21.(10  19.34 

1    o          '         "           " 

n5f.,  Sept.25  348     0  .54.5,25.0 
17()4  .Jiiii    15     12  37  39.0  50.4 

433. S 

435.0 

1.00 

+  .048 

_  0     1  49.4 
+  4  43  47.2 

40.0 
50.2 

-i-9.5 
(-17.4) 

_3.4 
(-V.0) 
44,4  1 
-0.5   \ 

-(-25.7 

1.05 

+  .013 

17(!8,  Dec.  27     31   2ti  52.0  32.0 
Dfc.  30     31   20  45. S  38.0 

12  15  35.0 
12  14  65.4 

30.0 
55.9 

419.4 
4   7.8 

413.0 

1.02 

+  .045 

1709,  .Inn.  15     31   22     7.7:55.8 
Jiin    10     31  22  23.4  11.1 

li   14  2(;.0: 
12   14  30.3 

29.7 
37.0 

411.9: 
412.3 

— 3.7: 
—0.7 

.Inn    20     31   24     0.0:43.7 

12   15  19.0 

18.« 

422.9: 

40.2   \ 

412.5 

1.01 

+  .049 

.Inn.  21     31   24  33.8  14.1 
Jan.  22     31   25     4.7  :47.7 
Jan.  23     31   25  28.5  24.2 

. ^ 

12  15  31.8 

30.8 

419.7 

41.0 

12  15  45.7 

44.7 

417. 0: 

41.0 

12  10    7.5 

:59.( 

,  +  ^■•^■- 

47.9  J 

AVlu-rc  no  declination  has  been  observe!  Ll.e  observed  corrections  m  right  ascen- 
sion have  been  changed  to  conretions  in  h.ngitnde  on  the  hypothesis  that  the 
theoretical  latitude  is  correct.     The  approximate  formula  is 

, .         (Vx  cos  h        , 

U-=     .     ,,,  where 
sin  jL 

cos  /;  =  sin  f  cos  a,  f  being  the  obliquity. 

DISCUSSION  OF  TIIK  MODKUN  OnSKIlVATION'S. 
TtciUctxon  of  the  PiiblUhed  Fesnlts  of  Olmrmilons  lo  a  Uniforvi  Sijstem. 
We  have  now  to  discuss  the  great  mass  of  observations  made  at  the  pvincipal 
observatories  of  the  world  since  the  discovery  of  the  planet  by  Herschel,  m  1 .81 
To  make  all  the  data  of  reduction  rigorously  homogeneous  and  uniform,  it  would 
be  necessarv  to  completely  re-reduce   the  greater  part  of  the  observations  maue 
befop.  1850   using  the  modern  values  of  the  constants  of  reduction   and  to  com- 
pare each  observation  separately  with  the  geocentric  place  deduced  from  the  pro- 
visional theorv.     Such  a  reduction  and  comparison  would  be  extremely  desirable. 
Their  execution  would,  however,  involve  an  amount  of  labor  for  greater  than  it  is 
now  possible  for  the  author  to  bestow  upon  the  problem.     A^  c  must,  therefore 
adopt  the  reductions  which  have  been  already  made,  applying  such  systematic 
corrections  for  reduction  to  a  uniform  system  of  star  places  as  we  have  the  means 
readily  to  determine.     No  reduced  places  are  employed  unless  we  can  find  data 
for  some  more  or  less  accurate  determination  of  these  -^7'=^7';;' '^  "f^;  "^^^J 
necessitates  the  rejection  of  a  great  mass  of  observations  made  at  the  mil  o    o W  - 
vatories  of  the  European  contin.Mit,  and  published  in   the  A.rononu.cJ.n  ^a?• 
riehten      We  still  have  the  following  rich  collection  of  materials  at  our  disposal . 


112  THE   OUBIT   OF   URANUS. 

1.  Observations  at  Greenwich,  1781  to  1872. 

2 Paris.  1802  to  1827,  and  1837  to  18G9. 

3 Kcinigsberg,  1813  to  1835. 

4 Vienna,  1822,  and  1827  to  1839. 

6 Speier,  1827-29. 

G Cambridge,  1828  to  1842. 

7 Edinburgh,  1836  to  1844. 

8 Berlin,  1838  to  1842. 

9 Vulkowa,  1841  and  1842. 

10 Washington,  18G1  to  1872. 

11 Leiden,  1863  to  1871. 

12 Santiago,  1854  and  1855. 

As  to  the  general  distribution  of  tliese  observations  in  time,  we  may  remark 
that  during  the  first  three  or  four  years  the  pUuiet  was  zealously  observed  at 
Greenwich.  Observations  then  began  gradually  to  fall  off  until  1798,  in  which 
year  we  find  but  one.  From  this  time  until  1814  only  one  or  two  observations 
were  made  at  each  opposition.  They  become  a  little  more  numerous,  until  1829, 
when  there  is  a  sudden  increase.  Few  interruptions  have  occurred  since.  With 
regard  to  the  other  observatories  it  may  be  said  that  from  1802  until  1830  there  is 
a  gradual  increase  in  the  number  of  observations,  and  that  since  the  latter  year 
the  number  of  observations  is  entirely  satisfactory. 

A  great  number  of  the  observations  were  reduced  with  the  star  places  of  the 
Tabula:  liegioniontana',  and  the  entire  Paris  series  arc  reduced  with  the  star 
positions  of  Le  Verrier,  given  in  his  '■'■Anmilcti  de  V  Obaervaloire  Imperial  de  Paris" 
Tome  II.  As  a  preliminary  to  the  discussion  of  the  systematic  corrections  to  the 
principal  published  reductions,  I  have  prepared  the  following  table,  showing  the 
corrections  which  must  be  applied  to  the  places  of  the  equatorial  fundamental 
stars  in  the  above  catalogues  to  reduce  them  to  the  adopted  standard,  namely.  Dr. 
Gould's  coast  survey  list  in  right  ascension,  and  Auwcrs'  standard  in  declination. 

In  the  table  of  right  ascensions  tlie  first  column  after  name  of  the  star  gives  the 
annual  variation  of  that  co-ordinate  for  the  epoch  1860.0,  as  derived  from  Le 
Terrier's  tables  of  right  ascensions  just  cited.  Next  we  have  the  corrcct'on  to  this 
annual  variation,  expressed  in  units  of  the  fourth  place  of  decimals,  ti  reduce  it  to 
that  given  in  the  "  Star  Tables  of  (he  American  EjJiemeris"  the  positions  in  which 
are  founded  on  Dr.  Gould's  Catalogue.  The  fourth  column  gives  the  correction  to 
the  right  ascensions  of  Lo  Verrier  for  18G0,  in  hundredths  of  a  second  of  time. 
Subtracting  from  this  column  sixth-tenths  of  the  preceding,  we  have  the  corre- 
sponding corrections  for  1800.  The  last  four  columns  give  the  corresponding  num- 
bers for  the  right  ascensions  of  the  Tabulai  RegiomontaiivR. 

The  table  of  declinations  shows,  for  different  epofihs,  the  corrections  necessary  to 
reduce  the  tabular  positions  to  those  given  by  Auwers  in  his  paper  on  the  declina- 
tions of  the  fundamental  stars 


THE  ORBIT   OF   URANUS. 


118 


• 

I.  RiaiiT  Ascensions. 

Corrections  to 

Ann.  var.  of  _ 

Corrections  to 

Date.                ann.  var. 

IStiU.           A 

nn.  var. 

R.  A. 

B.  A. 

Tah.  Keg. 

18B0.          Ann.  var. 

B.  A. 

18(iO. 

R.  A. 

1800. 

18(iO. 

1800. 

_ 

a  Andromeda), 

8 

+3.0844 

+  14 

+  2 

_  6 

3.0840 

+  18 

+  8 

—  3 

y  Pcgasi, 

3.0801 

+   15 

+  3 

_  G 

3.0824 

_-  8 

—  9 

—  4 

a  Arietis, 

3.3C44 

+  12 

0 

_  1 

3.3036 

+  20 

+  6 

—  6 

o  Coti, 

3.1266 

+   10 

+  5 

_  1 

3.1261 

+  9 

+  7 

+  2 

a  Tauri, 

3.4346 

_     1 

_  1 

0 

3.4335 

+  10 

+  T 

+  1 

/I  Orionis, 

2.8191 

+     G 

+  2 

_  2 

2.8800 

+  3 

+  5 

+  ^ 

3  Tauri, 

3.1871 

_     4 

+   1 

+  3 

3.7888 

_21 

—  4 

+  9 

a  Orionis, 

3.2460 

+     5 

+  ^ 

+   1 

3.2464 

+   1 

+  3 

+  2 

a  Gcminovani, 

3.8409 

—     ', 

0 

+   1 

3.8386 

+  21 

+  18 

+  5 
0 

a  Cauis  Min. 

3.1  i02 

+     '^ 

+   9 

+  6 

3.1455 

+  12 

+  T 

fl  Gcminorum, 

3.6828 

+     5 

+  3 

0 

3.6807 

+26 

+  13 

—  3 

a  Hvdroj, 

2.9485 

+   12 

+  fi 

—  1 

2.9469 

+28 

+  16 

—  1 

a  Leoui.s, 

3.2030 

+   15 

+  6 

_  3 

3.2014 

+  31 

+  13 

—  6 

p  Lcoiiis, 

3.0054 

+   11 

+  ^ 

—  3 

3.0640 

+25 

+  12 

—  3 

a  Virginis, 

3.1495 

+  20 

+  5 

K 

3.1497 

+  18 

+  2 

—  9 

a  Bootis, 

2.7325 

+  16 

+  3 

—  1 

2.7327 

+  14 

+  5 

—  3 

o'  Libra;, 

3.3044 

+     6 

+  3 

_  I 

3.3074 

_24 

—  6 

+  8 

—  5 
+  3 

—  3 

—  5 

—  5 

—  4 
+  2 

—  5 

^-  3 

—  2 
_  8 
—12 

—  6 
—55 

—.01 

a  CoroniB, 

2.5318 

+  12 

+  1 

—  6 

2.5373 

+  17 

+  5 

a  Scrpcntis, 

2.9488 

+     8 

+  * 

—  I 

2.9513 

—17 

—  7 

—  4 

a  Seorpii, 

3.6654 

+  16 

+  -1 

_  6 

3.6672 

•  Hcrculis, 

2.1322 

+     T 

+  2 

_  2 

2.1319 

+  9 

0 

a  Ophiuchi, 
a,  Lyra;, 
y  Aquilffi, 

2.1808 
2.0312 
2.8520 

+  10 
+     2 
+     9 

0 
—  1 

+  2 

_  fl 
_  2 

_  3 

2.1183 
2.0305 
2.8546 

+34 

+  9 
—17 

+  15 

+   1 
—  8 

a  A(inila!, 
3  Aquilaj, 

2.9281 
2.9466 

+     * 
+     5 

+   1 
+  3 

_  1 
0 

2.9281 
2.9496 

+  4 

—25 

—  3 
—12 

o'  Capriconii, 

3.3338 

+     * 

+  2 

0 

3.3349 

■      1 

—  6 

a  Aquarii, 
a  I'iiscis  Aust 

3.0829 
3.3311 

+  13 

+     8 

+  ^ 

_  4 
_  1 

3.0822 
3.3326 

+-n 
-  7 

—16 

a  Pogasi, 
Sum, 
Mean, 

2.9828 

+   11 

+  251 

+8.5 

0 

+  81 
+  .02 

—  1 
—72 
7    — 024 

2.9830 

+  9 
+210 

+7.0 

1 

—  1 

+71 
+  .024 

1 

15      Uar.  1S73. 


114 


THE    ORBIT   OP   URANUS. 


■ 

II.    DEC1>1NAT10((8. 

Corrections  to  Taliu 

»  Regiomontani 

Correellons  to  Le  Verrier. 

1780. 

1800. 

1820. 

1840. 

1820. 

1840. 

// 

n 

ir 

If 

// 

// 

a  Andromcdfc, 

-fO.3 

+0.2 

+0.1 

0.0 

+0.3 

0.0 

y  Pegasi, 

—0.2 

+0.1 

+0.4 

+C.3 

+0.1 

+0.3 

a  Arictis, 

—0.3 

0.0 

+  0.3 

+0.5 

+0.3 

+0.3 

a  Ccti, 

—1.0 

—0.1 

+0.7 

+  1.G 

+0.1 

+0.7 

a  Taiiri, 

0.0 

+0.1 

+0.1 

+0.2 

+0.1 

0.0 

|3  Orionis, 

—0.5 

+  0.1 

+0.C. 

+  1.3 

+0.3 

+0.7 

3  Tauri, 

—0.5 

0.0 

+0.f. 

+  1.1 

+0.2 

+0.5 

a  Orionis, 

—1.3 

— O.f. 

0.0 

+0.G 

—0.2 

+0.3 

a  aemiuornin, 

—0.8 

—0.5 

—0  1 

+0.3 

+0.5 

+  1.0 

a  C<>!!:3  Min. 

—0.3 

0.0 

+0.3 

+0.G 

+0.2 

+0.5 

3  Qcminorum, 

—0.3 

0.0 

+0.3 

+0.G 

+0.2 

+0.5 

a  HydroB, 

—0.2 

+0.3 

+0.7 

+  1.3 

+0.3 

+0.7 

a  Lconis, 

-{-0.3 

+0.4 

+0.4 

+0.5 

+0.4 

+0.5 

0  Lconis, 

+0.2 

+0.2 

+0.2 

+0.3 

+0.2 

+0.3 

a  A'irginis, 

+0.3 

+0.G 

+  1.1 

+  1.5 

+0.5 

+0.8 

a  Bootis, 

+0.4 

+0.4 

+0.3 

+0.3 

+0.4 

+0.4 

a?  LibriB, 

+0.5 

+0.5 

+0.G 

+0.G 

+0.5 

+0.fi 

a  CoroniB, 

+0.7 

+0.(5 

+0.4 

+0.3 

+0.4 

+0.4 

a  Serpentis, 

+0.2 

+0.G 

+  1.1 

+  1.5 

+0.5 

+0.9 

a  Scorpii, 

+0.1 

+o.r, 

+  1.0 

+  1.4 

+0.4 

+0.7 

a  llerculis, 

+0.6 

+0.8 

+  1.0 

+  1.3 

+0.5 

+0.7 

a  Opbiuchi, 

+0.4 

+0.5 

+0.5 

+  0.G 

+0.4 

+0.G 

a  Lyns, 

+0.7 

+0.8 

+  0.0 

+  1.0 

+0.4 

+  0.4 

y  AquiliB, 

0.0 

+0.3 

+0.G 

+  1.0 

+0.2 

+0.4 

a  Aquiloe, 

+0.1 

+0.4 

+0.7 

+1.0 

+0.1 

+0.4 

|3  AquiliB, 

+0.1 

+0.fi 

+  1.3 

+  1.7 

+  0.5 

+0.8 

»*  Caprieonii, 

+0.2 

+0.0 

+  1.G 

+2.3 

+0.6 

+  1.1 

a  Aquarii, 

—0.5 

+0.1 

+0.8 

+  1.5 

+0,5 

+  1.0 

a  Piscia  Aust. 

+1.0 

+  3.4 

+3.9 

+5.3 

+1.7 

+2.3 

a  Pegasi, 

+0.4 

+0.3 

+0.2 

+0.1 

+0.3 

+0.2 

Mean 

+0.03 

+0.3G 

+0.C9 

+  1.03 

+0.3G 

+0.60 

1 


THE   ORBIT   OP   URANUS. 


115 


The  coxTection  to  the  reductions  to  apparent  place  given  in  the  Tabulae  Rcgio- 
montansc  on  account  of  the  correction  to  the  constant  of  Nutation  is;— 

In  right  ascension : 
— 0".46  sin  ft  —  0".l8  sin  Q  sin  a  tan  S  —  0".24  sin  Q  cos  a  tan  8. 

In  declination: 
— 0"18  sin  a  cos  a  +  0.24  cos  Q  sin  a. 

The  terms  which  contain  tan  S  as  a  factor  may  be  entirely  neglected,  as  they  are 
small,  periodic,  and  contain  tan  h  as  a  factor  which  is  sometimes  positive  and 
sometimes  negative.     I  shall  also  neglect  tlic  corrections  in  declination,  as  their 

sum  is  sensibly 

0".21  sin  (a  -  8) 

the  effect  of  which  will  generally  be  confounded  with  the  accidental  errors  of 
observation. 

The  only  correction  we  shall  apply  on  account  of  nutation  is,  therefore, 

^a  =  — 0\()30  sin  Si. 
The  values  of  this  expression  at  the  dates  when  it  is  zero,  a  maximum,  or  a 
minimum,  are  as  follows : — 


1778.5 

1783.1 

1787.7 

1792.4 

1797.0 

1801.7 

1806.3 

1811.0 

1815.6 

1820.3 


—.03 

.00 
+.03 

.00 
—.03 

.00 
+.03 

.00 
—.03 

.00 


1820.3 
1825.0 
1829.6 
1834.3 
183S.9 
1843.6 
1848.2 
1852.9 
1857.5 


.00 
+.03 

.00 
-.03 

.00 
+.03 

.00 
—.03 

.00 


Having  adopted  this  system  of  standard  positions,  we  may  adopt  two  ways  of 
reducing  the  observations  to  it.  One  is  to  compare  the  positions  of  the  stars 
adopted  in  the  published  reductions  with  the  standard,  and  apply  the  mean  differ- 
ence to  the  reduced  place  of  the  planet.  Another  is  to  make  a  similar  com- 
narison  of  the  standard  cata1.-ue  with  the  positions  of  the  fundamental  stars 
which  have  been  deduced  fr-  m  observations  by  a  system  of  reduction  uniform 
with  that  employed  in  reducing  Uie  observations  of  the  planet,  and  to  regard  the 
mean  difference  as  a  correction  appU  :.ble  to  all  the  positions  of  the  planet.  If 
the  standard  catalogue  and  the  observations  are  both  free  fram  systematic  error, 
the  results  obtained  in  these  two  ways  should  be  substantially  identical.  1  hese 
are  however,  conditions  which  we  cannot  expect  to  find  fulfilled.  In  the  follow- 
in-^  discussions  I  have  sometimes  used  one,  sometimes  the  other,  and  sometimes 


116 


THE   ORBIT   OP   URANUS. 


Mr 


combined  both,  the  choice  being  determined  by  circumstances.  We  shall  con- 
sider the  different  series  of  observations  in  succession. 

Greeincicli  Ohaeivatii/ns  from  1781  to  1830, 

These  observations  are  completely  reduced  by  Airy  and  comp;.red  with  Bou- 
vard's  Tables,  in  the  work  licduction  of  the  Observations  of  Planets  made  at  the 
Royal  Observatory.,  Greenwich,  from  1750  to  1830,  London,  1845,  The  con- 
cluded positions  given  in  this  work  depend  mainly  on  the  star  places  of  the 
TabuliE  Regiomontanic,  both  in  right  ascension  and  declination.  If  we  consider 
the  first  four  oppositions — 1781-1785 — as  forming  a  single  group  of  which  the 
mean  epoch  is  1783,  wo  find  that  the  general  correction  to  the  TabultE  Ilegio- 
montanse  for  this  epoch  is 

In  right  ascension,     — 0'.030 ; 

In  declination,  +0".08. 

If,  on  the  other  hand,  we  consider  only  the  particular  stars  compared  with  Uranus, 
the  result  will  be  a  little  different.  The  number  of  times  each  of  the  fundamental 
stars  has  been  compared  with  Uranus,  and  the  correction  in  right  ascension  cor- 
responding to  each  star,  are  nearly  as  follows : — 


a  Arictis,         JV  = 

=  2 

Cor.  = 

-0.09 

iVX  c  = 

—  .18 

a  Tauri, 

2 

—.01 

—  .02 

y  Pcgasi, 

2 

-.03 

—  .06 

(i  Tauri, 

19 

+.13 

+2.47 

a  Orionis, 

33 

+.02 

+0.66 

a  Canis  Minoris, 

33 

—.02 

—0.66 

/3  Geminorum, 

34 

—.07 

-2.38 

a  Leonis, 

7 

—.11 

—  .77 

/i  Leonis, 

2 

—.07 

—  .14 

The  mean  correction  from  these  data  comes  out  — 0".008,  differing  by  0\022  from 
the  general  mean  correction.  Our  choice  between  the  two  (corrections  depends 
on  whether  we  are  to  consider  the  relative  positions  of  the  Tabultc  Regiomontamr, 
or  those  of  the  standard  catalogue,  as  nearest  the  truth  at  the  epoch  1783,  and 
particularly  upon  whether  we  arc  to  consider  the  large  correction  to  the  proper 
motion  of  (3  Tauri  as  real.  In  the  absence  of  exact  data  for  settling  this  ques- 
tion, the  mean  of  the  two  results,  or  — 0'.020,  has  been  adopted. 

A  similar  anomaly  is  exhibited  by  the  declinations.  It  is  probable  that  the 
declinations  of  Uranus  during  this  period  mainly  depend  on  stars  in  the  first 
twelve  liours  in  right  ascension,  for  which  the  mean  correction  is  about  — 0".,'50 
instead  of  +0".08.  I  have  adopted  — ()".16.  Changing  these  corrections  to  lon- 
gitude and  latitude,  we  have,  during  the  period  1781-1786: — 

Correction  to  observed  longitude,     =  — 0".30  ; 
Correction  to  observed  latitude,  — 0  .19. 


TllK   ORBIT    OP   URANUS. 


117 


Dtuing  the  years  1788-1798  the  above  systematic  iliffeience  in  right  ascension 
does  not  appear.     The  most  probable  correction  seems  to  be 


A8  =  0".00. 
Mat.  =  —()".  10. 


Aa=— 0'.02"j; 
Wliencc  A  long.  =  —  0".  34 ; 

Between  the  years  1800  and  1823  the  stars  used  for  comparison  are  so  widely 
scattered  tliat  I  consider  it  sate  to  apply  only  the  general  mean  correction  for  the 
epoch  1813,  which  is 

Aa  =  —  '.005 ;  Ac^  =  +0".()6. 

Whence  A  long.  =  0".00 ;  A  hit.  =  -f  0  .GG. 

From  1825  to  1830  more  than  half  the  weight  of  the  right  ascf^nsion  comes 
upon  the  stars  a,  (3,  and  y  AquiUv,  the  mean  correction  to  wliich,  during  tliis 
interval,  is  — 0'.035.  The  general  mean  correction  at  this  epocli  is  -|-0'.002.  I 
think  the  right  ascensions  of  these  three  stars  in  the  Tabultc  Kegiomontana;  are 
really  too  great  at  this  epoch  by  the  ehtire  diff'ereiue  of  these  results.  We  may, 
in  fact,  hereafter  regard  the  positions  of  the  standard  catalogue  as  sufficiently 
accurate,     'llie  mean  corrections  to  be  applied  will  llien  be 

A'(  =  — 0\017;  A5  =  +0".83. 

Whence  A  long,  =•  — O'.O.J  ;  A  hit.  =  -f<'"-^^- 

From  the  year  1831  until  the  present  time  the  Greeiuvich  observations  are  regu- 
larly reduced  in  the  several  annual  volumes  of  observations.  But  a  reduction  of 
the  observations  from  1831  to  1835,  executed  by  Mr.  Hugh  Breen,  is  given  in  an 
appendix  to  the  volume  for  the  year  18G4.  The  results  here  given  differ  from 
those  published  by  Pond  in  the  several  annual  volumes  for  tlie  same  interval.  The 
right  ascensions  are  altered  only  by  a])plying  the  constant  correction  — 0".030, 
which  is  found  necessary  to  reduce  Pond's  right  ascensions  to  those  of  the  Tabula; 
Regiomontana".  This  correction  I  have  verified.  The  mean  correction  to  reduce 
the  right  ascensions  of  the  Tabuhr  Regiomontana;  to  oiir  standard  is  at  this  time 
-|-0\005.  On  the  other  liand,  when  we  compare  the  concluded  right  ascensions 
of  stars  within  six  hours  of  Uranus,  as  given  by  Pond  in  the  (ireenwicli  observa- 
tions for  1834,  with  our  standard,  we  find  a  mean  correction  of  — ".034  to  reduce 
his  positions  to  the  standard,  wliieh  implies  a  correction  — ".004  to  Breen's  reduc- 
tion. The  two  residts  being  -|-'.005  and  — ".004,  I  have  applied  no  correction 
whatever. 

In  the  paper  in  question  the  declinations  are  comi>letely  re-reduced,  using  im- 
proved data  of  reduction,  but,  so  far  as  I  see,  making  no  changes  in  Pond's 
method.  The  results  diff"er  strikingly  from  those  of  Pond,  and  suggest  the 
desirableness  of  a  complete  re-examination  of  all  Pond's  determinations  of  decli- 
nation. Having  no  catalogue  of  observed  declinations  of  standard  stars  reduced 
in  this  same  way,  we  cannot  directly  determine  the  systematic  correction  to  the 
declinations.  I  therefore  proceed  as  follows:  A  comparison  of  Pond's  observed 
declinations  of  standard  stars  with  Aiiwers'  normal  catalogue  show  that  the  former 
require  the  following  corrections  near  the  parallel  of  Uranus : 


118 


THE   ORBIT  OF   URANUS. 


In  1831  —  1".42; 
1834  —  2.10. 

Then  comparing  Airy's  reduced  declinations  of  Uranus  with  Pond's,  we  find  tlie 
following  mean  differences: 

In  1831,  Airy  —  I'ond  =  —  3".  18 
1834,  —  3  .50. 

To  reduce  Airy  to  Auwers  we  must  there  apply  to  the  declinations 

In  1831  +  1".76 
1834  +  1  .40. 

I  have  regarded  the  correction  -j-  1".60  as  applicable  throughout  the  period  in 
question. 

1836-72. 

During  this  interval  the  corrections  in  right  ascension  have  been  derived  by  the 
following  two  sets  of  comparisons:  (1)  A  comparison  of  the  several  collected  six 
and  seven  ycai  catalogues  with  Gould's  standard,  from  which  it  appears  that  they 
require  the  following  general  corrections  in  right  ascension : 

Six  year  catalogue  of  1840  +  0'.047 

Six  year  catalogue  of  1845  -f  0  .002 

Seven  year  catalogue  of  1860  -f  0  .003 

Seven  year  catalogue  of  1864  -\-  0  .022 

(2)  A  comparison  of  the  corrections  applied  to  the  right  ascensions  of  the  indi- 
vidual years  to  reduce  them  to  the  standard  of  the  catalogue,  as  given  in  the 
introduction  to  each  catalogue.  The  sum  of  these  two  numbers  gives  the  correc- 
tions for  each  year. 

A  sliglitly  different  method  is  to  regard  the  above  correction  for  each  catalogue 
as  applicable  to  all  right  ascensions  which  depend  fundamentally  upon  that  cata- 
logue. I  have  sometimes  combined  both  methods  so  as  to  derive  what  seemed  to 
be  the  most  probable  result,  and  sometimes  used  but  one. 

The  corrections  to  the  declinations  during  tlie  interval  in  qnestion  have  been 
derived  from  Auwers'  "  Tafeln  zur  Reduction  der  DccUnationen  verxcJiicdener 
Stentverzeichnime  aiif  ein  Fundamentdxyfttem"  Astronomische  Nachncli/en,  No. 
1536.  These  tables  include  the  Greenwich  seven  year  catalogue  for  1860,  when 
the  correction  corresponding  to  the  declination  of  Uranus  is  about  -|-  0".45.  The 
corrections  for  the  previous  catalogues  vary  between  0".35  and  0".68.  The  cor- 
rection corresponding  to  the  interval  1861-67  has  been  derived  by  a  direct  com- 
parison with  Auwers'  declinations,  and  the  result  is  -|-  0".44,  agreeing  with  the 
two  preceding  catalogues.  But,  on  making  a  similar  comparison  with  the  annual 
catalogue  for  1S69,  a  considerable  change  was  found,  the  correction  being  —  0".17, 
a  change  of  more  than  half  a  second.  I  shall  use  this  correction  for  and  after  the 
beginning  of  1868,  as  the  change  is  probably  due  to  the  introduction  of  a  new 
constant  of  refraction  in  the  reduction  of  the  observations  for  1868  and  subse- 
quent years. 


i 


THE   ORBIT   OP   URANUS. 


119 


Cambridge, 
An  extended  scries  of  planetary  observations  was  eommcnccd  here  by  Professor 
Airy,  in  1827.  Tlie  series  was  continued  by  him  and  Professor  C'hallis,  his  suc- 
cessor, until  1842.  During  the  first  three  or  four  years  the  combined  right  ascen- 
sions depend  on  a  few  special  stars,  and  mainly  on  a^  Capricorni.  Taking  the 
mean  correction  to  the  adopted  right  ascensions  of  the  stars  actually  compared  as 
they  are  given  in  the  introduction  to  each  annual  volume,  giving  to  each  star  a 
weight  proportional  to  the  number  of  comparisons,  the  following  corrections  are 

deduced : 

1828  —  OMO 

1829-31     —0.10 

1832-37     —  O.in. 

In  the  introduction  to  the  volume  for  1838  it  is  stated  that  the  adopted  right 
ascensions  are  diminished  by  the  average  amount  of  ()'.()83,  wliich  would  still 
leave  a  correction  of  —  ()M07.     Actual  comparisons  in  two  subsequent  years  give 

1840,  A«  =  — 0'.087 

1842,  -  0  .069. 

Although  the  positions  deduced  from  each  year's  work  were  adopted  for  clock 

correction" the  year  following,  without  any  change  of  equinox,  it  seems  that  there 

was,  eifcctivcly,  a  progressive  change  of  about  0\0l  annually  in  the  equmox  as 

"  No*"  declinations  were  observed  until  1830.  On  comparing  the  declinations 
deduced  from  several  years'  work  with  Auwers,  it  was  evident  that  the  correction 
increased  with  the  polar  distance  of  the  star.  The  law  of  increase  could  be  well 
enough  represented  by  supposing  the  correction  proportional  to  N.  1 .  D.  i  bus  the 
following  corrections  were  deduced  in  three  different  years. 

1".78  X  N.  P.  D.  jn^egrees 
100 


1834,     h  dec.  =  — 


1840, 
1842, 


1".00  X  N.  P.  D.  in  degrees 


100 
1".03  X  N.  P.  D.  in  degrees 


100 


From  which  the  correction  for  other  years  was  deduced  by  interpolation.  But, 
on  applying  these  corrections,  the  results  were  found  systematically  difterent  from 
those  of  other  observatories,  and  on  referring  to  Auwers'  corrections  to  Airy  s 
Cambridge  Catalogue,  it  appeared  that  the  mural  circle  required  a  large  correction 
near  the  declination  of  Uranus  during  this  period.  The  above  results  were  there- 
fore  altered  so  as  to  conform  as  nearly  as  practicable  to  Auwers'  law. 

EiUnhtirgJi. 

In  reducing  the  observations  of  1836  Henderson  uses  the  right  ««^">«;°"«  °f 
theTabulffi  Ilegiomontana^,  to  which  the  general  correction  is  at  this  epoch  +  .UU1. 


199 


THE   OIIBIT  OF  URANUS. 


But,  if  wc  take  only  the  stars  near  Uranus,  with  which  the  latter  was  necessarily 
most  frequently  compared,  the  corrections  will  be  negative.  C'omparing  the  con- 
cluded positions  of  the  otars  from  a  Serpentis  through  0''  to  fj  Orionis,  we  Hud  tho 
following  mean  corrections : 

In  right  ascension,   —  0'.012;  in  declination,  — 0".09. 

In  subsequent  years  it  is  stated  that  the  adopted  positions  of  clock  stars  used 
each  year  ore  derived  from  the  right  ascensions  observed  at  Greenwich,  Cam- 
bridge, and  Edinburgli,  during  the  year  or  the  two  years  preceding,  without  any 
statement  whether  corrections  were  applied  for  ditfercnce  of  equinoxes.  In  some 
subsequent  years  the  following  corrc-ctions  are  deduced,  sometimes  from  the  adopted 
and  sometimes  from  tho  concluded  positions  : 

1837,     Aa  =      O'.OOO ; 

1840,     Aa  = +0.015;     A  Dee.  =      O'.OO; 

1844,     Aa  =  +0  .070 ;     A  Dec.  =  +0  .45). 

Far  18. 

All  the  positions  of  planets  given  by  Lc  Verricr,  in  his  '■'■Aniiahs  tie  VOhscrvn- 
toire  Impcmd  de  Paris:  C>i/<6/-f«/t«M«"depcnd  both  in  right  ascension  and  N.  P.  D. 
on  his  adopted  positions  of  fundamental  stars,  the  corrections  to  which  iiuve 
already  been  given.  As  the  corrections  to  the  in<lividual  star  places  used  by 
Le  Verrier  are  not  generally  of  a  systematic  character,  the  general  mean  correc- 
tion is  employed,  which  is:  — 

In  right  ascension  —  0'.024  +  0'.085  T, 
In  declination         +0".12    +  1".207; 

T  being  the  fraction  of  a  century  after  1800. 

In  1854  a  new  and  larger  catalogue  was  introduced,  and  for  this  and  the  follow- 
ing years  the  correction  in  declination  is  derived  from  Auwers'  tables. 

A  summary  of  the  adopted  corrections  after  1830,  as  deduced  from  the  pre- 
ceding comparisons  and  discussions,  is  given  in  the  following  table: — 


Table  (if  A 

DOPTED  SySTE.MATIO  CcttUEOTIONS. 

Year. 

Greenwich. 

Paris. 

Kijiiigsberg. 

Berlin. 

Cambridge. 

Eiiinbiirgh. 

Aa 

AS 

Aa 

A8 

Aa 

Aj 

Aa 

AS 

Aa 

AS 

Aa 

AS 

s 

n 

8 

n 

B 

// 

8 

II 

8 

8 

1830 



.00 

.... 

+0.9 

-.16 

—1.7 

.00 

1831 

.00 

+  1.8 

.00 

+0.5 

—.02 

+d.'9 

.... 

+0.9 

-.16 

—1.6 

1832 

.... 

+  l.t 

•  • .  • 

—.02 

+0.9 

.... 

.... 

—.19 

—1.5 

.... 

1833 

+  1.6 

—.04 

.... 

—1.4 

.... 

1834 



+  1.4 





.... 

—.05 

.... 

—1.3 

.... 

1835 

+  1.2 

.... 

.   •   •    • 

—.05 

.... 

•  at* 

—1.2 

.... 

1836 

—.04 

+  1.0 

+.01 

+0.6 

—.02 

+  1.0 

—.04     +1.0 

.... 

—1.0 

—.01 

0 

1837 

—.03 

+0.9 

.... 

>    •    •    < 

—.03      .... 

—0.8 

.00 

0 

1838 

—.04 

+0.8 

.... 

.... 

—.02  1   . . . . 

—.11 

—0.5 

+.01 

0 

1839 

.... 

+0.T 

.... 

.... 

....    1+1.0 

—.01 

.... 

—.10 

—0.3 

+  .01 

0 

1840 

.... 

+0.(! 

.... 

.... 

1 

.00 

—.09 

—0.2 

+  .02 

+0.1 

1841 



+0.5 

.... 



1 
....    j   .... 

+.01 

—.08 

—0.2 

—.04 

+0.2 

' 


TUE  Oil  BIT  OF  UUANVB. 


191 


Table  or  Adoi>tkd  Systematic  Cobbections.— Co«/tnu»(/. 

Vuttr. 

Oraonwluli. 

I'irU. 

Kilnlgslwrg. 

U«rliil. 

Caiulirlilgs. 

K<lliil>urgh. 

1842 
1843 
1844 

1845 
184(; 
1847 

1848 
1849-53 
1854-55 
l85(J-(;o 

1801 
i802-f.5 
l8(!li 
18(17 
1808 
18(i0 
1870 
1871 
1872 

As 
8 

—.02 

+  .07 

+  .00 

+  .08 

+  .04 

+  .05 

+  .05 

.00 

.00 

—.01 

—.01 

.00 

.00 

.00 

.00 

.00 

+  .01 

+  .01 

+  .02 

AS 

II 

+0.4 
+  0.4 
+0.4 
+  0.4 
+  0.4 
+  0.4 
+0.4 
+  0.4 
+  0.5 
+0.5 
+0.4 
+0.4 
+0.4 
+0.4 
—0.2 
—0.2 
—0.2 
—0.2 
U.2 

A. 

+  "01 
+  .01 
+  .01 
+  .01 
+  .02 
+  .02 
+  .02 
+  .02 
+  .02 
+  .02 
-1  .03 
+  .03 
+  .03 
+  .03 
+  .03 
+.03 
• . . . 

AS 

II 

+0.0 
+  0.('. 
+  0.0 
+0.0 
+  0.0 
+  0  0 
+0.0 
+0.0 
+0.2 
+  0.2 
+  0.2 
+  0.2 
+  0.2 

+o.a 

+0.2 
+  0.2 

Ao 
H 

AS 

Aa 
8 

+.03 
+.04 
+  .04 
+  .04 
+.03 
+.02 
+  .01 
.00 
—.01 

At 
n 

+1.1 
+  1.1 
+1.1 
+  1.1 
+  11 
+1.1 
+  1.2 

+  1.2 
+  1.2 

Aa 

8 

—.07 

Sillll 
.00 

.  .  •  • 
.... 

AS 

II 

— o.:i 

in(fo 
-j  0.0 

A»           At 

8 

—  02     +0.8 
+  .07     +0.4 
+  .07  1  ^  0.6 

Wusliington 

.00    —0.5 
.00     +1.1 
.00  j  +1.1 
.00     +1.2 
.00     +0.0 
.00     +0.4 
.00     -4-0.4 
.00     -i-0.4 

Applyiiij?  the  prrroding  corroctions  to  tho  positions  of  tho  planet  as  orijjinally 
Tediux'd  and  publishi'd,  we  liavc  a  sciii-s  of  observed  positions  as  nearly  homo- 
geneous as  it  is  possible  to  make  them  with  the  means  now  at  our  command.  Tlio 
next  step  in  order  will  be  tiio  computation  of  the  geocentric  place  of  the  planet 
from  the  provisional  theory  for  the  moment  of  every  observation,  to  be  compared 
with  the  results  of  the  latter.  The  complete  execution  of  this  labor,  ab  initio,  is, 
however,  at  i)rcsent  impracticable,  and  it  is  proposed  to  diminish  it  by  making  use 
of  the  published  comparisons  with  the  older  tables.  Tliis  can  be  done  without 
danger  of  serious  error,  and  with  all  the  more  case  that  owing  to  the  great  distance 
of  Uranus  the  errors  of  the  solar  tables  are,  for  the  most  part,  without  appreciable 
effect  upon  tho  computed  geocentric  place  of  tho  planet.  Tlie  method  of  making 
the  comparison  is  different  with  different  series  of  observations,  and  each  scries 
must  therefore  be  described  and  discussed  separately.  The  general  plan,  how- 
over,  has  been  to  replace  observed  and  computed  absolute  positions  by  observed 
and  computed  corrections  to  the  geocentric  positions  deduced  from  Bouvard's 
Tables.  To  carry  out  this  plan  it  is  necessary  to  have  at  our  disposal  an  ephemeris 
botli  of  the  heliocentric  and  geocentric  positions  derived  from  these  tables.  Tlie 
corrections  to  the  latter  given  by  the  observations  are  then  given  by  direct  com- 
])arison.  To  obtain  the  corrections  given  by  the  provisional  theory,  the  heliocentric 
longitudes,  latitudes,  and  radii  vectores  given  by  tliat  theory  arc  interpolated  to 
the  dates  of  the  heliocentric  ephemeris  from  Botivard's  Tables,  and  compared  with 
that  ephemeris.  The  differences  are  tli(>n  chang(Ml  to  differences  of  geocentric 
place  by  the  usual  differential  formuhr,  and  thus  tlio  cornn'tions  given  by  theory 
are  derived.     The  difference  between  the  two  sets  of  corrections  is  the  difference 

10      May,  1873. 


122 


T  II  K   0  11  H  IT   OF  U  II  A  N  U  S. 


between  the  provisional  theory  niul   observation,     A  roti«b'ns(>(l  ■nmmary  of  the 
results  for  euch  of  the  priiieipal  tieries  of  observations  is  here  presented. 

Oirciivlch,  nsl-lSHO. 

In  Airy's  rediietions,  ahcady  refcrvi-d  to,  we  liave  given  for  the  moment  of  each 
individual  obNervation  u  helioet^iitiie  place  computed  from  IJouvard's  Tables,  nnd 
the  geocentric  longitudes  and  latitudes  thenee  deduced.  The  observed  right 
ftscensions  and  declinations  are  then  changed  to  longitudes  nnd  latitudes,  nnd  thi; 
apparent  error  of  the  tables  thence  deduced.  The  means  of  these  errors  nre  taken 
for  groups  of  observations,  and  expressed  in  terms  of  tho  errors  of  heliocentric 
longitude,  radius  vector,  and  latitude.  The  mode  in  which  these  means  have  been 
treated  is  fully  shown  in  the  following  table.  The  first  colunm  gives  tho  mean 
date  of  each  individual  group  of  observations.  The  ne.\t  three  give  the  mean 
excesses  of  the  co-ordinates  interpolated  from  the  heliocentric  ephemeris,  p.  100, 
and  corrected  for  solar  nutation,  over  those  printed  in  the  "Compututions  of  tabular 
place,  etc.,"  in  the  Greenwich  reductions.  In  tlie  fifth  column  these  corrections 
nre  changed  to  corrections  of  geocentric  longitude.  In  the  next  two  columns  we 
have  tlie  mean  corrections  to  Bouvard's  geocentric  places  given  by  observation.  It 
is  the  negative  of  the  mean  error  of  tabular  place  printed  in  the  "  Reductions," 
corrected  by  the  numbers  ulready  given  to  reduce  the  star  places  to  ii  nniforiu 
system.  Then  we  have  the  difference  between  these  two  sets  of  corrections,  or, 
the  mean  correction  to  the  geocentric  place  of  the  provisional  theory  as  given  by 
observation.  Lastly,  we  have  the  differential  coefficients  for  expressing  tlie  errors 
of  geocentric  in  terms  of  the  errors  of  heliocentric  co-ordinates  taken  without 
change  from  the  Greenwich  volume. 


From  ProTinional  Theory. 

From  ObiiiTTationa. 

~^ 

rorri-i'tlnn  to 

Mean  Date. 

Corroction  to  latmlar  poaitlou  in 
UreHUwich  Kuductiona. 

Correction  to 

I'lOV. 

'i'l»«ory. 

dl 

69 

Long. 

W. 

Ut. 

Geoo. 

Oeoo. 

H«l. 

Noof 

(1(«>0. 

R.  V. 

long. 

long. 

lut. 

Olw.  1    long. 

1. 

// 

If 

ff 

If 

II 

1791,  Oct. 

10 

—  7.8 

•    •    •    • 

+1.1 

—  7.5 

—  (5.4 

+  2.0 

4 

+  1.1 

+  1.5 

1.01 

—  9 

Nov. 

13 

—  7.5 

+1.1 

—  7.5 

—  4.9 

—   1.0 

4 

+2.0 

—  2.7 

1.04 

—  6 

Dec. 

27 

—  7.3 

—  i59 

+  1.0 

—  7.7 

—  5.(i 

—  2.3 

5 

+2.1 

—   3.3 

1.05 

+  1 

1782,  J  Ml. 

31 

—  7.4 

+0.9 

—  8.0 

—  4.9 

—  1.1 

4 

+3.1 

—  2.0 

1.04 

+  fi 

Mar. 

4 

—  G.G 

+0.9 

-  7.1 

—  4.(i 

—  0.0 

4 

+2.5 

—   1.5 

1.01 

+  9 

Oct. 

10 

—  5.0 

+0.4 

—  5.3 

-3.C 

—  4.0 

4 

+  1.7 

-    4,4    1.01 

—  9 

Nov 

2(i 

—  5.3 

+  0.3 

—  5.3 

—  0.0 

—  5.0 

3 

—0.7 

~  5.3 

1.04 

—  5 

1783,  .Ian. 

11 

—  5.0 

+0.3 

—  5.4 

—  3.9 

—  3.7 

3 

+  1.4 

—  4  0 

1.05 

4-3 

Feb. 
Oct. 

24 
10 

-  4.4 
—  3.5 

+0.2 
—0.3 

—  48 

—  3.2 

—  2.2 

—  0.3 

—  2.4 

—  1.0 

8 
2 

+2,0 

—  '..6 

—  0.7 

1.02 
1. 00 

+  9 
—10 

+2.9 

Nov. 

1 

—  3.2 

—0.3 

—  3.0 

—  1.1 

—  2.3 

2 

+  1.9 

—  2.0 

1.02 

—  9 

Dec. 

15 

—  3.3 

—  i2i 

—0.4 

—  3.4 

—  2.2 

—  3.8 

3 

+  1.2 

—  3.4 

1.05 

—  3 

1784,  .Ian. 

29 

—  3.1 

.  •  •  • 

—0.4 

—  3.4 

—  2  4 

—  2.4 

3 

+  1.0 

—  2.0 

1.05 

+  5 

Mar. 
Oct. 

12 
30 

—  2.6 

—  1.8 

.... 

—0.5 
—0.8 

—  2.9 

—  1.5 

—  0.3 

—  0.7 

—  4.3 
+   1.4 

3 
2 

+2.0 

—  3.8 

1.01 
1.02 

+  9 
—  9 

+0.8 

+  2.2 

Dec. 

14 

—  1.5 

—  i28 

—0.9 

—  1.5 

—  2.4 

+  2.7 

2 

—0.9 

+  3.0 

1.05 

—  4 

Til  13   OUUIT    OF   UUANUa. 


198 


Kiuiii  rruvltluual  Tliuury. 

Kruiu  OUarTtttiuua. 

M««u  Datii. 

CurrHCtliin  tn  tnlmliir  pnaltlou  In 
(li'ouiiwiuli  iilmirvittioiiii. 

Carrsatlon  to 

Correilloii  to 
Piov.  riieoiy. 

vl 

6-/. 

dl 

Lo(i. 

Lat. 
n 

Ohoo. 

Oeoo. 

llol. 

No.of 

n«oo. 

H«l. 

Long. 

R.  V. 

long. 

long. 

lut. 

UU. 

long. 

lat. 

// 

It 

II 

II 

II 

1785,  Jhii.   KI 

—  1.2 

—  1.0 

—  1.3 

—  1.4 

—  1.4 

2 

—0.1 

—  0.4 

1.05 

+  8 

KrI..    1.1 

—   1.0 

—  1.1 

—   1.3 

—  1.2  i 

—   1.1 

2 

+  0.1 

0.0 

1.04 

+  ' 

Mm.  20 

Nov.     H 

—  O.fJ 
+  0.2 



—1.1 
—1.5 

—  0.9  ;  —  2.1 
+  0.5  '  +   1.2 

—  5.2 
» 

—  1.7 

2 
2 

—1.2 

—  4.1 

1.01 
1.02 

+10 
—  9 

+0.7 

—  0.2 

1788,  Mur.  l.'l 
Oct.   2« 

+  a.l 
+  3.4 

— 'i87 

—3.2 
—3.0 

+   1.8     +   4.4  1 

—  4.9 

—  2.2 

3 
2 

+2^ 

—  1.7 

1.03 
1.00 

+  9 
—10 

+  3.9 

+  3.0 

—0.9 

+  1.4 

178'J,Jan.  18 

+  4.2 

-'224 

—3.7 

+  4.5 

+  1U.4 

—14.5 

1 

(+6.9)  (—10.8)'  1.00 

—  1 

Apr.     8 
Oct.   31 

+  ^■1 

+  4.8 

—3.8 
—4.2 

+  4.2 

+  5.4 

+  9.4 
+  4.7 

—  3.7 

—  1.1 

4 

2 

+  5.2 

+  0.1 

1.01 

+  10 
—10 

—0.7 

+  3.1 

1.00 

1790,  Juu.  24 

+  4.9 

4.4 

+  5.2 

+  3.1* 

—  2.6 

2 

—1.4 

+   1.8 

1.00 

—  1 

Nov.     5 

+  6.1 

-  254 

—4.8 

+  5.8 

+  5.0 

—  2.3 

3 

—0.8 

+  2.5 

1.00 

—  10 

1791,  Jan.  29 

+  5.1 

—  258 

—6.0 

+  5.4     +  4.5 

—  3.8 

3 

—0.9 

+   1.2    1.06 

0 

Apr.  14 

+  5.0 

.... 

—5.2 

+  4.5     +  2.7 

—  4.8 

1 

—1.8 

+  0.4  !  1.01 

+  10 

Nov.  10 

+  5.0 

-  234 

—5.4 

+  5.0     +  5.7 

—  4.0 

2 

+0.1 

~+   l74  1  1. 00 

—10 

1792,  Feb.    5 
Nov.  15 

-f  5.2 
+  5.0 

—5.0 
—0.0 

+  5.5 
+  fi.2 

+  4.2 
+  3.7 

—  4.0 

—  2.5 

1 
3 

—1.3 

+   1.6 

1.00 
1.00 

0 
-10 

—2.5 

+  3.5 

1793,  Feb.     8 

t  5.4 

—0.1 

+  5.7 

+  8.6 

—  5.0 

2 

+  2.8 

+  1.1  i  1.06 

0 

Nov.  14 

+  5.0 

—  135 

— G.4 

+  5.9 

+  10.4 

—  6.3 

1 

+4.5 

+  0.1 

0.99 

—10 

1794,  Fob.  15 

+  5.7 

—  100 

—0.6 

+  0.0 

+  7.3 

—  0.3 

2 

+  L3 

+  0.2 

1.00 

0 

Nov.  19 

+  5.4 

—     07 

—0.7 

+  5.0 

+  4.1 

—  5.1 

4 

—1.5 

+   1.6  f  1.00 

—10 

1795,  Feb.  20 

Nov.  29 

+  5.9 
+  5.4 

—  40 

—  13 

—0.9 
—7.0 

+  0.3 
+  5.5 

+   0.0 
+  3.9 

—  8.0 

—  3.4 

3 
2 

—0.3 

—  1.1 

1.06 
1.00 

0 
—10 

—1.6 

+  3.6 

179fi,  Feb.  24 

+  5.3 

-       8 

—7.2 

+  6.6 

+  4.5 

—  4.5 

2 

—1.1 

+  2.7 

1.06 

0 

1797,  Feb.  27 
1800,  Mur.  14 
1814,  Mfty  22 

+  5.1 
+  4.0 
—  1.2 

+     C2 
+  224 
+  217 

—7.5 

—8.4 
—2.2 

+  5.4 
+  4.9 
—  1.3 

+  4.7 
+  4.9 
+   1.1 

—  4.2 

—  8.7 
+  0.2 

3 
2 

2 

^0/7 

+  3.3 
—  0.3 

1.00 
1.00 
1.00 

0 
0 
0 

0.0 

+2.4 

~+  2.4 

1815,  May  25 

—  1.1 

+  214 

—1.5 

—  1.2 

+   1.4 

—  1.0 

4 

+2.6 

+  0.5 

1.00 

0 

1818,  June  10 

—  0.5 

+  413 

—0.8 

—  0.5 

—  1.1 

+  4.8 

2 

—0.6 

+  5.6 

1.00 

0 

1819,  Juno  14 

—  0.4 

+  483 

+  1.5 

—  0.4 

—  1.4 

+  3.7 

4 

—1.0 

+  2.2 

1.00 

0 

1820,  June  Ifi 

—  0  1 

+  498 

+2.2 

—  0.1 

—  2.0 

+  4.0 

2 

—2.5 

+  1.8 

1.06 

0 

1823,  July     1 

0.0 

+  538 

+4.4 

0.0 

+  0.2 

+  5.0 

4 

+  0.2 

+  0.6 

1.05 

0 

1825,  July  11 

—  2.4 

+  507 

+5.8 

—  2.5 

—  4.8 

+  7.8 

2 

—2.3 

+  2.0 

1.05 

0 

1820,  July  l(i 

—  4.5 

+  582 

+0.0 

—  4.7 

—  3.0 

+  8.3 

4 

+  1.1 

+  1.7 

1.05 

0 

1827,  July  20 

—  0.0 

+  02( 

+7.2 

—  0.9 

—  7.2 

+  9.6 

6 

—0.3 

+  2.4 

1.05 

0 

1828,  July  23 

—  9.8 

+  710 

+7.9 

—  10.3 

—  0.9 

+  0.9 

3 

+3.4 

—  1.0 

1.05 

0 

1829,  Aug.    7 

—13.4 

+  83.'i 

+8.5 

—13.7 

—16.3 

+  10.0 

14 

—2.6 

+   1.5 

1.08 

+  ? 

Oct.     4 

—14.2 

+  SO,") 

+8.0 

—12.5 

—13.0 

+  9.9 

8 

—0.5 

+  1.3 

1.02 

+  ^ 

1830,  July  30 

—17.8 

+  97C 

+9.0 

—18.8 

—21.4 

+11.1 

3 

—2.6 

+  2.1 

Aug.  29 

—18.2 

+  98!> 

+9.1 

—17.it 

—20.0 

+10.2 

5 

—2.1 

-4-   1.1 

Sept.  20 

—18.5 

+  99i 

+9.1 

—17.2 

—19.1 

+  9.1 

12 

—1.9 

0.0 

Oct.    14 

—  18.9 

+100f 

+9.1 

—10.8 

—17.3 

+  9.9 

11 

—0.5 

+  0.8 

Nov.  13 

—19.3 

+  100J 

+9.2 

—16.8 

—17.8 

-\    a.  3 

8 

—1.0 

+  0.1 

124 


THE   OUBIT   OF   URANUS. 


Parh,  1801-1827. 
A  complete  reduction  of  this  series  is  found  in  Le  V'errier's  Anindvs  lie  I'Obser- 
valoifc  Imj}erud  (Ic  Purin,  OhserontioHK,  tome  1.  No  comparison  with  any  ephemeris 
is  given  here,  nor  i«  there  any  complete  ephemeris  to  compare  them  with.  A 
complete  geocentric  ephemeris  was  therefore  computed  from  the  provisional  theory 
for  the  principal  groups  of  the  Paris  observations.  Tlic  individual  observations 
being  compared  with  it,  the  resulting  mean  corrections  arc  given  in  the  following 
table : 


Menn  date. 

Aa 

AS 

N. 

Mcnn  datfi. 

Aa 

AS 

N. 

1801,  ^larch  24, 

—".02 

+  1".2 

2 

1813,  May  20, 

+M9 

+r.8 

6 

1802,  April  1, 

+  .08 

+0.6 

13 

1814,  May  27, 

+  .21 

+0.8 

4 

1805,  April  22, 

+  .10 

+2  .2 

13 

1815,  May  24, 

—  .02 

+2  .2 

5 

1806,  April  17, 

—  .01 

-1.6 

5 

1816,  June  1, 

—  .01 

+0.8 

7 

1807,  April  28, 

+  .17 

+0.4 

16 

1817,  June  5, 

—  .08 

+  1.6 

5 

1808,  April  28, 

+  .02 

+  1.4 

6 

1818,  June  7, 

+  .12 

+2  .2 

9 

1809,  May  5, 

+  .20 

+0.1 

9 

1819,  June  18, 

—  .07 

+  1.8 

7 

1810,  April  30, 

+  .22 

+2  .6 

16 

1820,  June  20, 

-.20 

—2.4 

8.5 

1811,  Fcbr'y  18, 

+  .21 

+2  .2 

3 

1821,  June  22, 

+  .05 

+  1.0 

5 

1811,  May  17, 

+  .14 

+2.6 

8 

1823,  July  18, 

+  .02 

+  1.8 

5 

1812,  Febr'y  1(5, 

+  .28 

2 

1824,  July  13, 

+  .04 

0.0 

7 

1812,  May  10, 

+  .16 

+3  .0 

6 

1827,  July  25, 

—  .05 

+0  .6 

5 

1813,  Fcbr'y  25, 

+  .44 

3 

Total  number  of  observations  in  right  ascension,  175. 

The  observations  in  this  series  exhibit  numbers  of  discordances  of  that  class 
which  leave  the  astronomer  in  doubt  whether  the  observation  should  be  retained 
or  rejected.  This  remark  applios  more  especially  to  the  declinations.  If  we  de- 
termine the  probable  error  of  an  observation  in  declination  by  the  condition  that 
it  is  that  amount  which  the  error  fulls  short  of  as  often  as  it  exceeds,  it  is  found 
to  be  about  2".  Then,  if  the  errors  followed  the  commonly  assumed  law  of  proba- 
bility, only  about  one  in  six  of  the  errors  should  exceed  4",  and  one  in  twenty- 
three  6".  But  errors  of  these  magnitudes  are  much  more  numerous,  the  deviations 
often  amounting  to  six  or  eight  seconds.  I  nave  rejected  only  a  few  in  which  the 
discordances  approached  10". 

BessiTs  IConiifgherg  Obxcrrntions,  1814-1835. 
T  have  made  a  complete  re-reduction  of  the  right  ascensions  of  this  important 
series,  and  of  most  of  the  declinations.  In  order  to  avoid  the  necessity  of  ap|)ly- 
ing  systematic  corrections.  Dr.  Gould's  right  ascensions  and  Dr.  Anwers'  declina- 
tions were  used  througho)it  in  these  reductions.  In  this  work  a  selection  of 
the  fundamental  stars  observed  by  Bessel  was  made  for  each  observation  of  the 
planet,  to  be  used  for  clock  error.  These  were  chosen  so  tliat  the  mean  of  their 
right  ascensions  and  declinations  should  be  as  near  as  practicable  to  those  of 
Uranus,  a  condition,  however,  which  coidd  not  generally  be  fulfilled  for  the  decli- 
nations, owing  to  the  southern  position  of  the  planet.     Bessel's  instrumental  cor- 


THE   OUBIT   OP   URANUS. 


126 


roctions  were  applied  to  his  observed  times  of  transit  over  the  mean  wire,  and  the 
resulting  ti.ne  was  employed  as  that  of  transit.  Each  time,  compared  with  the 
computed  right  ascension  of  the  star  gave  a  value  of  the  clock  correction,  which 
was  reduced  to  the  time  of  transit  of  the  planet  by  the  known  daily  rate.  It  the 
instrumental  errors  were  always  accurately  determined,  the  mean  of  these  clock 
corrections  would  be  used  to  obtain  the  right  ascension  of  Lranus.  But  it  was 
frequently  found  that  the  clock  error  varied  systematically  with  the  declination  of 
the  star,  so  that  it  was  deemed  advisable  to  add  to  the  clock  correction  a  term 
varying  as  the  simple  declination,  which  was  deduced  from  all  the  stars,  and  used 
to  reduce  the  correction  to  the  parallel  of  Uranus. 

It  was  intended  to  give  the  results  of  this  reduction  for  each  observation,  but 
on  comparing  the  results  with  those  of  Fleming  in  the  AstronomiscJie  Xachnchten, 
Band  30,  it  appeared  that  the  results  were  not  materially  better  than  his.  It 
does  not,  therefore,  seem  necessary  to  give  more  than  the  mean  results  for  each 

opposition.  . 

From  Bessel's  declinations,  with  the  old  Cary  circle,  I  was  unable  to  obtain  any 
satisfactory  results,  owing,  apparently,  to  a  wai.t  of  knowh-dge  of  some  pernl.ar.ty 
„f  tlie  instnmunt.  Fleming's  reductions  were  therefore  adopted,  ihey  are 
designated  by  the  letter  F  in  the  following  list. 

Mean  Corrections  to  the  Prormonal  Eplicmeris  ijiven  ly  Bo^scVs  Oh^^crvalhns  at 

Koii  l<jxher<),  1 8  U- 1 829. 
AS  K  I         Moan  ilftto. 

-]-2".i)F      9    1822,  June  24, 
_|_1  .8/.'     11     1823,  July  4, 
+  1.2/''    11  \  1824,  JulyC), 
+2  .3/'      8  !  1825,  July  K), 
4-4  :.\F    13  i  1826,  July  18, 
+4.1  4  I  1828,  July  25, 

+  1  .5         5  !  1829,  Aug.  1, 

Total  numbers  of  observations,  103. 

Results  of  Ohscrvat!n»s  at  various  Ohserratorlrs,  from  1827  to  1829  lurlush-c. 
During  these  three  years  we  have,  besides  the  observations  already  quoted,  the 

following:  — 

1  Observations  bv  Schwcrd,  at  Sp.-ier,  of  which  the  originals  are  given  in 
Astrouo,ulsvJ,r  Ikohal.Um,jen  amjrsicti;  av/ <1rr  Sler.irarte  drs  Kiiul.,1.  J.>/z>ums  lu 
SjH;,rr  vou  F  M.  Mwerd,  Speyer,  1S29-30,  an.l  of  whi.h  the  reduced  results 
are  found  iu  tlie  Astronomiscbe  Naehricliten,  Ba-.id  8,  S.  2(14.  ^       ,    .       , 

2.  The  series  by  Airy,  at  Cambridge,  commenced  iu  1828,  and  found  in  the 
Cambridge  Observations. 

3.  Littrow's  Vienna  Observations,  found  in  the  fiist  series  of  Annahn  drr  A.  A. 

Stennvartc  In  Wlcn. 

The  mean  corrections  to  the  ].rovI.  ional  epbenieris  given  by  these  series  are 
shown  iu  the  following  table.     The  observations  have  been  divided  in  the  usual 


Menu  ilal?. 

1814,  May  2\ 

1815,  May  25, 
181(),  May  27, 
IS17,  June  0, 
1818,  June  8, 

1820,  June  21, 

1821,  June  23, 


Aa 
+M1 
+  .13 
+  .()') 
+  .13 
+  .02 
+  .02 
+  .12 


Aa 
+M0 

—  .05 
+  .01 
+  .01 

—  .01 

—  .15 
—.10 


+r'.8 
-1  .()/<' 

—  1  .{)F 
-2  .4/'' 
—3  .OF 
-3  J)F 
-1  .OF 


N. 


J 


126 


TUE   ORBIT   OP    URAXUS. 


way  into  groups  of  about  a  month  each,  and  the  mean  date  and  mean  correction 
found  for  each  group.  The  Paris  ond  Konigsberg  results  are  repeated  for  tlie 
sake  of  clearness.  The  small  figuies  show,  as  usual,  the  number  of  observations 
employed  in  forming  the  mean. 


Aa 

Ad 

Date. 

Observatory. 

Original.     Corrected. 

1827, 

July  22, 

Speier, 

— 0M6,  — 0M4 

July  25, 

Paris, 

_()  .0:3.  —0  .05 

+0".53 

September  15, 

Vienna, 

— 0.11,,— 0.10 

0  .0,. 

October  U, 

Vienna, 

—0.18,   —0.17 

-2.2, 

1828, 

July  25, 

Kiinigsberg, 

—0.15,   —0.15 

-3.5, 

July  29, 

Vienna, 

—0.24,   —0.20 

-1.4, 

August  14, 

Vienna, 

—0  .13,,,  —0  .09 

+  1  .1,0 

August  27, 

Speier, 

—0.10,   —0.09 

September  18, 

Vienna, 

—0.03,  +0.01 

+  1.0, 

September  25, 

Cambridge, 

_0.05,  —O.IG 

October  17, 

Vienna, 

—0  .13,,  —0  .09 

0.0„ 

October  17, 

Cambridge, 

—0  .02,,,  —0.12 

1829, 

August  1, 

Konigsbcrg, 

_0.1()„  —0.10 

—  1.0 

August  6, 

Cambridge, 

+0.11,  -0.08 

-1.1.7 

August  28, 

Speier, 

_0.04,  —0.04 

Soptembor  23, 

Cambridge, 

+0.21,0+0.05 

November  6, 

Cambridge, 

+0.25,   +0.09 

Obnervations  from  1830  io  1872. 
Since  the  year  1830  heliocentric  and  geocentric  ephcmerides  of  Uranus  com- 
puted from  Bouvard's  Tables  are  at  our  disposal.  We  make  use  of  those  in  the 
Berlin  Astronomisches  Jahrbuch  for  the  years  1830  to  1833,  and  of  those  in  the 
Nautical  Almanac  from  1834  forward.  The  system  of  comparison  is  the  same 
as  that  already  explained.     That  is  to  say,  we  deduce  separately : 

(1)  Mean  corrections  to  the  geocentric  longitude  and  latitude  of  Uranus  hi  the 
ephereris  as  derived  from  observation. 

(2)  Mean  corrections  to  the  same,  given  by  the  provisional  theory,  as  derived 
from  a  comparison  of  the  heliocentric  positions  of  that  tiieory  with  the  heliocentric 
positions  in  the  ephemeris. 

Then  (1)  —  (2)  is  the  correction  to  the  provisional  theory  given  by  observation. 
The  process  of  forming  (1)  and  (2)  is  shown  quite  fully  in  the  following  jiages. 
Each  individual  printed  observation  was  first  compared  with  the  printed  ephemeris, 
and  a  correction  to  the  latter  was  thence  deduced.  AVhen  this  correction  was 
given  with  the  observations  themselves,  it  was  of  course  not  recomi)uted,  uidess  in 
some  doubtful  cases.  The  observations  were  then  divided  into  groups,  usually  of 
about  a  month  each,  and  coinciding  in  time  with  the  grouping  of  the  Greenwich 
results.  The  mean  of  the  dates  and  the  mean  of  the  corrections  were  then  taken 
separately  fir  each  group  and  each  observatory.     The  separate  results  arc  shown 


THE   ORBIT   OP  URANUS. 


127 


I 


J 


in  the  proper  columns  of  the  following  table,  under  the  head  "Mean  dates," 
Mean  cor.  in  R.  A.,  and  Mean  cor.  in  Dec.  These  means  are  those  given  by  the 
observations  as  printed,  without  the  application  of  the  systematic  corrections  on 
pages  120  and  121.  In  the  columns  "  Corrected  mean"  these  corrections  are  applied  ; 
this  column  would  therefore  exhibit  no  systematic  differences  between  the  results 
of  the  different  observatories,  unless  the  observations  of  Uranus  were  affected  by 
errors  different  from  those  which  affect  the  positions  of  the  fundamental  stars.  A 
careful  comparison  of  the  differences  in  various  parts  of  the  table  shows  that  this 
is  unfortunately  the  case.  A  weight  is  next  assigned  to  each  individual  result 
depending  on  the  number  of  observations,  the  general  sufficiency  of  the  data  of 
reduction,  the  mean  discordance  of  the  individual  observations,  and  the  quality  of 
the  instruments.  The  critical  reader  will  notice  a  lack  of  homogeneity  among  the 
weights  assigned,  of  which  I  shall  speak  presently.  The  mean  of  the  separate 
group-results  is  then  taken  with  regard  to  these  weights,  and  also  the  mean  of  tlie 
mean  dates,  using  for  the  latter  the  relative  weights  adopted  for  the  several  right 
ascensions.  Thus,  we  have  a  mean  result  derived  from  all  the  observations  for 
each  month,  or  other  group-period,  which  is  written  under  the  horizontal  lines. 

These  corrections  to  right  ascension  and  declination  are  next  changed  to  correc- 
tions of  longitude  and  latitude,  using  for  this  purpose  the  following  table,  which 
is  computed  from  the  formula;  of  Gauss : 

cos  E  =  sin  E  cos  a  sec  b  =  sin  e  cos  I  sec  5 

,  cos  /;  . . 


sin  jP  cos  5   ^ 

Al=: ,    -  Aa 

coso 


cos  b 

Ab=  —  cos  E  cos  5  Aa  +  sin  E  Ab. 
The  differential  coefficients  in  this  table  are  expressed  as  a  function  of  the  right 
ascension  of  Uranus  only,  which  may  be  done  because,  owing  to  the  small  iuclina- 
tiou  and  great  distance  of  the  planet,  its  geocentric  position  on  the  celestial  sphere 
is  never  more  than  about  2'  from  some  point  of  the  projection  of  its  heliocentric 
orbit.  The  coefficients  of  A«  are  multiplied  by  15,  that  the  right  ascension  may 
be  expressed  in  time. 


To  Convert  Eiuiors  of  Riqht  Asoe.nsio.v  and  Declination  op  Ukanus  into  errors  op 

LoNdlTUDE  AND  LATrfUIlE. 
When  the  Ilight  Asieiision  exceeds  12',  enter  with  R.  A  — 12^  aii.l  oh-iiige  the  signs  of  Uie  qnanlities 

('  b      ,  61 

6a        od 

■J.    A. 

0"     0"" 
10 
20 
30 
40 
60 

1       0 
10 
20 
30 
40 
60 

Logniithms  of 

61 

6a 

+13.8+ 
+13.9+ 

6b 
6a 

-5  97+ 
6.96 
6  95 
5.92 
6.87 
6.81 

-5.74+ 
5.66 
6.56 
5.45 
5.33 
5.20 

61 

6S 

6b 

6b 

61 

6a 

1.1386 
1.1387 
1.1388 
1.1389 
1.1390 
1.1391 

1.1392 
1.1393 
1.1394 
1.1394 
1.1395 
1.1396 

6h                   61          \     6b 
6a                  66          '    68 

—0.7761           ,      +9.6000           .       9.9626 
—0.7757  ""  ,1       +9.5996  ~  „       9.9626 
—0.7743  "~  '...       +9.5983  ~   ',,       9.9628 
—0.7720  ~  ';;      +9.5963  "~  f,'       9  9632 
—0.7687  —  ii      +9.5934  ~  T^l      9.9637 
—0.7643  ~  ll      +9.5896  _  ;,"       9  9645 

-0.7588         ,,„       +9.5849         .,       9,9653 
-0.7525  —  ''■]       +9  57114  ~  '{[       9.9662 
-0.7451  ~  I*      +9.573(1  ~  74       9  9673 
-0.7  65  ~  ,';      +9.56.-.6  ""  r*.       9.96^5 
-0.7J70  -,;  ;;       +9.5573  -  :;:       9.9698 
-0.71B2  ~},g      +9.5479  ^^^^      9.9711 

+0.40— 
0.40 
0.40 
0.40 
0.39 
0.38 

+0.38— 
0.3K 
0.37  ■ 
0.37 
0.3(i 
0.36 

+0.92+ 
-4  0.92+ 

128 


THE   ORBIT   OP   URANUS. 


To  CoNVEiiT  EiiRons  or  Uiqiit  Asci 

.\sioN  AND  Declinatki.n.  —  C'oiiliniicd. 

B.  A. 

LogarithniB  of 

61 

,5ft 

61 

O^ft 

()l 

eft 

5Z 

6b 

da 

6a 

^5 

dS 

6a 

6a 

'66 

+  0.93+ 

21'    0" 

1.1.395 

-0.7<i44      „q 
— 0.6915       \-., 
-(1.6773  -J;^- 
-0.6618       \'.Z 
-0.6450  -]f, 
-0.6266  Zw 

+  9.5375  _  J. 
+  9.5260       }.^* 
+  9.5134  -|^^ 
+9.4995  X  •'? 
+0.4843       Jf.2 
+9.4676  X}^I 

9.9725 

+13.8+ 

—5.06+ 

+0.34- 

10 

1.1, !95 

9.9740 

4,91 

0.33 

2(1 

1.1395 

9.9756 

4,75 

0.32 

30 

1.1395 

9.9772 

4,58 

0.31 

41) 

1.1395 

9.9788 

4,41 

0  30 

BO 

1.1394 

9.9804 

4,23 

0.29 

3     0 

1.1394 

— O.C069      niR 
_0..'.854  -j'i: 

—0.5368  —r;l^ 

-0.5094  -f'* 
-0.4795  Z^^ 

+9.4495       ,.„ 
+9.4297  -}^,l 
+9.4081  X^i? 
+  9.3844       -i' 
+  n.358(i       ^»° 
+  9.3302  _^J* 

9.9821 

+13.8+ 

—4,05+ 

+0.28— 

+0.96+ 

10 

1.1394 

9.9837 

3.86 

0.27 

20 

1.1.393 

9.9853 

3.66 

0.26 

3U 

1.1393 

9.9869 

8,45 

0.24 

40 

1.1392 

9.9883 

3.24 

0.23 

60 

1.1391 

9.9898 

3.U2 

0.21 

4     0 

1.1390 

-0.4468      „,„ 
-0.4109  _lll 
—0.3710  _fji 
-0.3267  -i^^ 
-0.2769      *i, 
-0.2198  ZZl 

+9.29S0       „.„ 
+9.2644  Zttt 
+9.2259       ^.^\ 
+9.1828       "1 
+9.1341       ,^»^ 
+9  0781X2^? 

99912 

+13.8+ 

—2.80+ 

+0.20— 

+0.98+ 

10 

1.1390 

9.9925 

2.58 

0.18 

20 

1.1389 

9.9938 

2.36 

0.17 

30 

1.1388 

9.!'9.n 

2.14 

0.15 

40 

1.1387 

9.:':.-U 

1.91 

0.13 

60 

1.1386 

9.y'  J 

1.67 

0.12 

5     0 

1.1385 

—0.1539       -„, 
-0.(1753  ~'l^. 
-9979     -,?,^ 
-9.854        };l 

—9.378         ■^^^ 

+9.0130      »-_ 
+8.9353  _^^J 
+  8.8397  _f^% 

+8.7157    |:^" 

+8.540    _}/■:' 
+8.240        •^^J^ 

9.997'i 

•  + 

—1.43+ 

+0.10— 

+0.99+ 

10 

1.1384 

9.9984 

1.19 

0.08 

20 

1.1383 

9.9990 

0.95 

0.07 

30 

1.1.382 

9.9994 

0.71 

0.05 

40 

1.1381 

9.9997 

0.^8 

0.03 

50 

1.1380 

9.9999 

0.24 

0.02 

6     0 

1.1379 

—  oc 

<X 

0 

+13.7+ 

0.00 

0.00 

+1,00+ 

10 

1.1378 

+9.378     jlo"; 
-(-9.«79     +fl 
+  9.854     +\l' 
+9.978     +^;* 
+0.0743  +^11 

-87157  +1^" 
«-^^"  +1240 

-89353  +95« 

9.9999 

—0.24+ 

—0.02+ 

20 

1.1377 

9.9997 

0.48 

0.03 

30 

1.1376 

9.9994 

0.71 

0.05 

40 

1.1375 

9.9990 

0.95 

0.07 

50 

1.1374 

9.9984 

1.18 

0.08 

7     0 

1.1373 

+02753  +6'»' 
+S:.3260  +'^, 
+0.3692  +t!^ 
+0.408S  +^«J 

—9.0130    ,.., 
-9.0781  X*" 
-9.1341  ±5^'* 
-9.1828  Xl^i 
-9.2259  X*2J 
-9.2644 +33««, 

9.9977 

+1?.7  + 

+1.42— 

—0.10+ 

■  1-0,99+ 

10 

1.1373 

9.9969 

1.65 

0.12 

20 

1.1372 

9.9959 

1.88 

0.13 

30 

1.1371 

9.9949 

2.11 

0.15 

40 

1.1371 

9.9938 

2.34 

0.17 

50 

1.1370 

9.9925 

2.56 

0.18 

8     0 

1.1370 

+0.4448    ,  „„-, 
+0,4773  X-^^ 
+0,.';071  J^l 
+0,5344  +^l 
+0.5597  +2:*,^ 
+0.6830  +2j'3 

-9.2990    ,  ,,„ 
-9.3302  +111 
-9.3586  Tf* 
-9.3844  +^^8 
-9.4(181  X^*J 
-9.4297  +\ll 

9.9912 

+13.7+ 

+2.78— 

—0.20+ 

+0.98+ 

10 

1.1370 

9.9898 

3.00 

0.21 

20 

1.1370 

9.9883 

3.22 

0.23 

30 

1.1370 

9.9869 

3.43 

0.24 

40 

1.1370 

9.9853 

3.03 

0.26 

50 

1.1370 

9  9837 

3.83 

0.27 

9     0 

1.1370 

+0.6044    ,  ,„ 
+0.6241  +  11 
+0.6425  +\f 
+0,6594  +    '^ 
+0.6749  +\'^, 
+0.6892  +\jl 

-9.4495    ,  ,„. 
—9.4676  t\% 
-9.4843  +  5^^ 
-9.4995  +j°2 
-9.5134  t"^ 
-9.6260  +111 

9.9821 

+16.7+ 

+4.02- 

—0.28+ 

+0.96+ 

10 

1.1370 

9.9804 

4.20 

0.29 

20 

1.1370 

9.9788 

4.38 

0.30 

30 

1.1371 

9.9772 

4.55 

0.31 

40 

1.1371 

9.9756 

4.72 

0.32 

50 

1.1372 

9.9740 

4.38 

0.33 

10     0 

10 

1.1373 
1.1374 

+  0,7023    ,  , 

+0.7434  X  ^/j 
+0.7510  +  ^5^ 

.  -9,5375    ,  J.. 
-9.5479  +^"* 
— 9.M73  X  81 
— 9.5(;66  X  74 
—9  5730  X  ,'I 
-9.5794  +  5^* 

9.9725 
9.9711 

+13.7+ 

+5.04— 
6.18 

-0.34+ 
0.35 

+  0,93+ 

20 

1.1376 

9  9698 

5.31 

0.36 

30 

1.1376 

9.9685 

5.43 

0.37 

40 

1.1377 

9.9673 

6.54 

0.37 

50 

1.1378 

9.9602 

6.64 

0.38 

11     0 

1.1379 

+0.7575    ,    ., 
+0.7(i32  +  " 
+0.7678  +  *° 
+0.7713  +  Z 
+0.7737  X  ^* 
+0.7754  +  ^ 

—9.5849    ,     .- 
—9.5896  X  QS 
-9.5934  X  on 
-9.5963  +  :^; 
-9.69.K3  X  i\ 
—9.6996  X     4 

9.9i;53 

+  13.7+ 

+5.72— 

-0.38+ 

+  0.92+ 

10 

1.1380 

9.9645 

5.79 

0.38 

20 

1.1.381 

9.9037 

6. SO 

(1.39 

30 

1.1382 

9,9032 

6.90 

0.39 

40 

1.1. J83 

9.9628 

6.94 

0.39 

60 

].13()5 

9.9626 

5.96 

0.40 

12     0 

1.1386 

+0.7761 

—9.6000 

9.9626 

+13.8+ 

+6.97- 

— «.40+ 

+0.92+ 

THE   ORBIT   OP   URANUS. 


129 


Wc  thus  have,  for  the  interval  occupied  by  each  group  of  observations,  a  mean 
correction  to  the  geocentric  longitude  and  latitude  of  the  planet  given  by  obser- 
vations, which  are  fountl  in  the  ninth  and  tenth  columns  of  the  table,  on  the  same 
horizontal  line  with  the  mean  corrections  in  right  ascension  and  declination  from 
which  they  arc  derived.  The  next  step  is  to  obtain  the  corresponding  corrections 
given  by  the  provisional  cphemeris. 

This  correction  has  been  first  obtained  for  every  twentieth  day  of  each  of  the 
forty-two  oppositions  included  in  the  table.  The  heliocentric  longitude,  latitude, 
and  radius  vector  were  interpolated  to  the  most  convenient  twenty-day' intervals', 
and  compared  with  the  corresponding  co-ordinates  in  the  heliocentric  eplicmcris.' 
This  cphemeris  was  of  course  the  one  corresponding  to  that  with  which  the 
observations  were  compared,  namely,  the  Berliner  Jahrb'uch  for  the  years  1830-33, 
and  the  Nautical  Almanac  for  subsequent  years.  These  comparisons  arc  fullj' 
given  at  the  end  of  this  chapter,  and  the  resulting  corrections  to  the  printed 
cphemeris  are  given  in  the  proper  columns  of  the  table. 

These  corrections  to  the  heliocentric  co-ordinates  were  then  changed  to  corrections 
of  geocentric  longitude  and  latitude  by  the  following  formula'.     Put 

»•',  the  projection  of  the  planet's  radius  vector  on  the  cclii»tic ; 

p',  the  projection  of  the  planet's  distance  from  the  earth  on  the  same  plane ; 

p,  this  distance  itself; 

X,6\  the  planet's  heliocentric  longitude  and  latitude ; 

Z,  the  sun's  geocentric  longitude ; 

Jl,  its  radius  vector; 

M,  the  modulus  of  the  common  logarithms; 

il,  if),  the  corrections  to  the  geocentric  longitude  and  latitude ; 

5p,  the  correction  to  the  common  logarithm  of  tlie  radius  vector 

Then 


61 


=    ^|l+5cos(Z-;i)| 

—    ^  s:n  (Z,  —  7.)  jf4-^, 
p  J/sm  1 


hX 


-\--r,  sin(/.  —  ?.)tim,33^i 

Ih  =     -';   I  1  -I-  ilf  tan  V  cos  (Z  -  ?.)  I  h^ 

r'Ul 

,„  tan  ;i  sui  (Z  —  X)  fU 

PP 

+  ''^1   {  1  +  ;'  cos  (/.  _  X)  I  sin  i3  -,,  ^P  ,.- 

The  last  tern?  in  hi  and  tl;e  last  two  terms  of  fb  have  been  omitted  in  the  com- 
l)utation,  as  they  scarcely  ever  exceed  a  few  hundredths  of  a  second. 

17       May,  1873. 


130 


THE   ORBIT   OP   URANUS. 


The  values  of  H  and  ib  are  printed  in  the  last  two  columns  of  the  table, 
formula  for  hb  might  have  contained  the  additional  term 

ib  =  sin  ISa 


The 


fiu  being  the  correction  to  the  obliquity  of  the  ecliptic  adopted  in  the  ephcmeris 
to  reduce  it  to  tliat  employed  in  tiie  provisional  theory.  This  correction  is,  how- 
ever, deferred  until  wc  conio  to  form  the  equations  of  condition. 

From  the  values  of  tl  and  hb  thus  obtained  we  are  to  find  the  mean  values 
during  each  group  of  observations.  If  these  quantities  varied  uniformly,  the 
proper  value  would  be  that  corresponding  to  the  mean  date  of  each  group.  But 
the  second  ditferences  arc  so  large  that  this  value  would  generally  be  in  error  by 
one-  or  two-tenths  of  a  second.  Owing  to  the  minuteness  of  this  difference,  it  has 
been  considered  that  when  tiie  mean  date  was  near  the  middle  of  a  twenty-day 
interval,  the  correction  fil  interpolated  to  that  date  without  regard  to  second 
differences  would  furnish  a  sufficient  approximation  to  the  required  mean  value  of 
H  during  an  interval  of  about  30  days.  In  other  case  the  value  of  fl  was  inter- 
polated to  5-day  intervals  through  the  period  of  each  group  of  observations,  and  tho 
mean  value  taken. 

During  the  years  '850-1863  the  sun's  longitude  employed  in  the  ephemeris 
required  a  gradually  .iicreasing  correction,  amounting  at  the  latter  date  to  about 
3".  A  small  correction  of  which  the  maximum  value  is  about  0".15  was  applied 
to  tl  to  reduce  it  to  the  value  it  would  have  had  if  Hansen's  tables  had  been 
employed. 

The  corrected  mean  values  of  II  and  !h  thus  obtained  arc  given  in  the  last  two 
columns  of  the  following  table,  being  inclosed  in  brackets  and  printed  innnediately 
above  the  values  of  Al  and  Ab  derived  from  observation. 

1  deem  it  proper  to  mention  that  the  mechanical  labor  of  constructing  these  tables 
of  comparisons,  in  the  manner  just  described,  was  in  great  part  performed  by  Dr. 
C.  li.  F.  Kampf,  who  was  enijdoyed  by  the  Smithsonian  Institution  to  assist  me  in 
the  work.  Before  using  it  I  subjected  the  whole  of  the  work  to  a  careful  revision, 
altering  especially  the  relative  weights  of  the  corrected  means  in  many  cases.  As 
the  assigned  weights  now  stand,  each  set  of  results  which  arc  combined  into  a 
i'  ^-j-  mean  has  its  own  unit  of  weight,  which  does  not  necessarily  coincide  with 
that  of  any  other  set.  The  use  of  a  uniform  scale  of  weights  through  this  scries 
of  observations,  and  the  assignment  to  every  final  mean  of  a  weight  equal  to  the 
sum  of  the  weights  of  the  quantities  whose  mean  was  taken,  would  have  led  to 
weights  in  many  cases  quite  fictitious,  owing  to  the  obvious  presence  of  systematic 
errors  in  the  residts.  For  tliis  reason  I  have  made  no  further  use  of  the  weisrhts 
found  in  this  table,  and  their  lack  of  homogcncousn(>ss  therefore  does  no  harm. 


lesss 


TUE   0U13IT   OF   URANUS. 


131 


T 


Mean  CoiuiiiCTioNS  to  tiik  Kpiiemkhis  ok  Uranvs  in  tiik  Bkiii-inku  Jaiiuducu  and  the 

Aaltical  Almanac. 


Obaervatory. 
[It.  A.  of 
Crauus.  ] 


Mean  dates. 


Obaerred  oarrections  in  R.A. 


Konip;sl)org, 
Cambridge, 

[20"  40'°] 

KOnipKbcrg, 
Cambridge, 

[20'  3G'°] 

Cnmbridgo, 
[20"  37'"] 

Cambridge, 
[20"  3(1'"] 

Cambridge, 
[20»  3'i"'] 

Grcenwieh, 
Cambridge, 

[20"  M'"} 

Greenwicli, 
Cambridge, 

[20"  l>-2'"] 

Grcenwieh, 
Cambridge, 

[20"  50"'] 


Greenwich, 
Konig.«berg, 
Camliridge, 
Vienna, 

[21"  U""] 

Cambridge, 
[21"  12"'] 

Cambridge, 
Vienna, 

[21"  0'"] 

Cambridge, 
A'ienna, 

[21"  lO"] 


1830 

July  29 

July  29 

July  29 

Aug.  12 

Aug.  25 

Aug.  19 

Sept.  19 

Oct.  17 

Xoir.  14 

1831 

Aug.  3 

A  ug.  H 


Mean. 


Aug.  G 

Sept.  7 

Sept.  15 

Sept.  11 


Xov. 
Oct. 


4 

20 


,s 
— 1.5(i 
—1.51 


-l.f.5 
-1.30 


-1.4f. 


—  1.34 


—1.30 


Oct.  31 

1832 

Aug.  9 

Aug.  10 

Aug.  15 

Aug.  3 

Aug.  12 

Sept.  12 


Get. 
Oct. 


0 
12 


Oct.        1 


Nov. 
Nov. 


Ifi 
9 


-1.72 
-1.70 


-l.f.7 
-1.00 


-1.4S 
-1.54 


-2.02 
-2.24 
-1.99 
-2.33 


-1.97 


-1.91 

-i.;io 


-1.S5 
-1.89 


Nov.     15 


No.  of 

obd. 


11 

9 


10 


10 


13 
15 


Corrected 


Observed  correotiona  in  Dec.     Corr.  to  Qeocentrlo 


Mean. 


8 

—1.50, 
— 1.C7, 

—1.00 

-1.05, 
—  1.52, 

—1.58 
-1.62, 


—1.50, 


-1.52, 


—1.12. 
—1.86, 

—1.79" 

—LOT, 
-1.70, 


—1.72 

-1.4S. 
—1.70. 

— 1.C3 


-2.02, 
-2.24; 
-2.18, 
-2.33, 

-2.19 
-2.1c,. 

-2.09,^ 
-1.90^ 

-2.01 

-2.04, 
-1.89, 

-2.02 


+4.0 


-I-3.G 


-f2.5 
-f3.5 
-f3.1 

-fl.7 
40.2 


-fl.9 


+  1.1 


No.  of   Corrected   Lougitude. 
obd.        ueau. 


n 


-1-4.0 


-f3.C 


-fl.S 

+5.3 
+  4.9 

+3.4, 
+  1.1 


4  2.5 


15         +2.8 


+  2.0 


Latitnde. 


[-19.2]  [+  9.5] 
—20.8      +10.3 


[-18.6]  [+  9..5] 
—20.8      +  9.1 

[-17.2] 
—21.2 

[-16.7] 
—19.5 

[-10.0] 
—19.8 


[—23.8]  [+10.0] 
—23.4      -j-11.1 


[—22.0]  [+10.0] 
—22.2  I   +11.8 


[— 20.7][+  9.8] 
—21.0      +10.8 


[—28.4]  [+10.3] 


29.2 

[—20.91 

— 28.8 


+  11.9 


[—25.8]  [+10.2] 
— 2G.G      +11.1 


[—25.2]  [+10.1] 
—27.4  .   +10.5 


132 


THE   0 11  BIT    OF   U  II A  N  U  S. 


Mean  Coukeutions  to  the  Kimiemeuis  or  Ukani-s.— C'o»<(Hwtrf 


Obmirvatory. 
[K.  A.  ur        Mean  dntei. 
Drauui.] 


ObBerr><d  corrvutfons  In  B.  A.    Oliserrvd  oorreotions  In  Deo. 


Muan.       No  of    Corrnclf il 
ubi4.    j     uiuaii. 


Oreciiwich, 

Kdriip[.sl)i'r)f, 

Cambridge, 

[21"  32™] 

Qroi'iiwich, 
Cuiulji'i(l|ru, 
Vieium, 

[21''  23"] 

Oroeiiwicli, 
Caiiiliriiijfe, 
Viemin, 

[21"  2G"] 

Ciimbriilije, 
Vieiiiiiv, 

[21"  2(i™] 


Cambridge, 
Vie:iiia, 

[21"  49"] 

Grpenwicli, 
Cambridge, 

[21"  45'"] 

Greenwich, 
Cambridge, 
Vienna, 

[21"  41'"] 

Cambridge, 
Viunna, 

[21"  41'"] 

Ciinibiidge, 
Vienna, 

[21"  4:{'"] 


Oreenwicli, 
Kiinigsbcrg, 
(Cambridge, 
Vienna, 

[22"  4'"] 


1833 

Aug. 

22 

Aug. 

12 

Aug. 

15 

Aug. 

15 

Sept. 

18 

Se|>t. 

1!» 

Sept. 

11 

Sept. 

1.S 

Oet. 

11 

Oct. 

12 

Oet. 

14 

Oet. 

12 

\ov. 
\ov. 


1« 

14 


Nov.    in 


1834 

Aug.     1 
Aug.    1; 


Aug. 

Sept. 
Sept. 


14 

10 
Hi 


Sept.    13 


Oct. 
Oct. 
Oct. 


14 

l(i 
20 


Oct.      17 


Xov. 
Nov. 


Hi 
12 


Xov.     14 


Dec. 
Dec. 

Dec. 


1835 


A  ug. 

Aug. 
Aug. 
Aug. 


17 
20 
14 

21 


Aug.     17 


R 

-2.57 
-2.57 
-2.40 


-2.3.3 
-2.33 


-2.25 
-2.20 
-2.37 


-2.10 
-2.24 


-2.58 
-3.07 


-3.00 
-2.00 


-2.74 
-2.(18 
-2.75 


-2.04 
-2.53 


-2.7! 

-2.55 


-3.25 
-3.31 
-3. 28 
-3.30 


M<'an. 


7 

5 

11 


5 
10 


4 

12 
10 


11 
4 


5 

18 


I'l 


4 

11 

20 

1 


H 

-2.57, 
-2.57. 
— 2.5'.i., 

—2.58 

-2.3.!, 
-2.52. 
-2.50, 

—2.47 

2. 25 

— 2.48J 

—2.43 

-2.35, 
o_28 

—2.33 


—3.11, 

—2.83 


—3.00, 
-2^ 

—2.88 

—2.74, 

—2.87, 


4        _ 


2.80, 
—2.83 

-2.83, 
-2.58, 

—2.77 


1        _ 


2.00, 
•  50' 


—1.4 
—1.5 


—0.8 
-f2.8 
-f2.0 


+  0.1 
-f2.2 


+    1.3 


1.3 
4.4 


4.4 
1.2 


3.1 
0.8 
1.3 


0.7 

2.8 


1.2 

2.1 


—2.88 


—3.25,  '  —  fi.O 
—3.31, 

—3.47,:—  4.7 
— 3.35„    _  8.8 

—3.39 


No.  of 
oIm. 


4 
H 

0 


4 

13 


11 
4 


5 

18 


10 

10 

4 


11 

8 


20 
1 


Cormotdd 
uieuii. 


+"•2, 
—  I  .5. 


—0.0 

H-0  8. 

+  1.2, 
+  1.2 

41-T. 


-1-1. 0 
—0.1 


Corr.  to  Geocentriu 
Lougltude.    Latitude. 


[—33.8] 
— 35.0 


[  +  10.8] 
-f  10.9 


[—32.2]  [+10.8] 
-33.4       -)-12.3 


[-31.2]  [+10.0] 
—33.0       +11.0 


[—30.3]   [  +  10.3] 

—0.1      I  —32.0    ,   +10.4 


—2.0 
—3.5 


—2.7 

—3.0. 
2  5 

—2.0 

-l-7,„ 
-2.1,„ 

—1.9 
—2.0,, 


i_[_40.0]  [  +  11.1] 
—39.7       +11.3 


[-30,0]  [  +  11.1] 
—40.5       +11.2 


[-37.5]  [+10.9] 
—39.5    i   +11.0 


■■'     [_3f,.7]  [+10.7] 
2.0        —38.7    ,    +11.3 


-2., 5. 
—  1.2i 

—2.4 


-5.4, 


-5.9., 


[—30.2]  [  +  10.5] 
—40.3       -f  11.5 


:^Aj[_47.1]  [+11.3] 
-5.9        —48.0       +11.8 


THE   OKU  IT  OF   URANUS. 


133 


Mean  Corhkctions  to  tub  Ki'iiKMKitm  ok  Uranus. — Continued. 


Olifler»iitory. 
[K.  A.  of 
Uiauua.] 


Canibridgo, 
Vii'iiiia, 

[22"  1'"] 

Oi'ot'nwicli, 
Cambridj^o, 

[21''  57'"  I 

Orcciiwicli, 
C'umlji'iilgc, 

[21"  57'"] 


Orecnwicli, 
Ciimbridge, 

[22"  21'"] 

Qi'ccnwirli. 

Ciunbridifo, 
I'Jiliidiui'^h, 
V^ieniia, 

[22"  20"] 

Greoinvic'i, 
Uaiiil)ridj?(', 
Kdiiiliiii'^li, 
Vii'iina, 

[22"  l(i">] 

Qrcpnw  it'll, 
CamhridfTc, 
Kdiribiii'>;li, 
Vienna, 

[22"  l;)'"] 

Green  wii'li, 
Camljrid^c, 
E)diid)iii'^li, 
V^ienna, 

[22"  12"'l 

Camhrid^o, 
Edinljurgii, 

[22"  i:{"'] 


QreiMitt'ii'h, 
[22"  40'°] 


Ub8Hrr«(l  uorraotions  in  R.A.  ,  Observed  correotiona  in  Deo.     Corr.  to  Oeocebtrio 


Mean  dates. 


1835 

Sept.     15 
Sept. 


Sept. 

Od. 
Oct. 

Oct. 


II 

15 

10 
17 


Xnv.  27 

Xov.  2() 

Nov.  2(1 

1836 

July  22 

.Inly  25 

July  2;j 


A  lift. 
Auf;. 
A..-. 
AiiK. 

AngT 

Sept. 
Sept. 
S.pt. 
St'pt. 


24 
!J0 
Hi 
1!» 
20 


Mean. 


— :M'? 
— y.;50 


—.3.27 

—a.  11 


-:!.2i 
-i.oo 


-.1.H0 

-;{.<!  0 


— .•i.7S 


No.  of 
obii. 


Corrected 
luuuii. 


Mean. 


Sept.    15 


Oct. 
Oct. 
Oct. 
Oct. 


12 
1(1 
15 
11 


Oct.      K 


Xov. 
Xov. 
Xov. 
Xov. 


i;i 

l:i 

17 

« 


Xov.     13 


Dec. 
Dec. 


12 
12 

12 


1837 

July     22 


-;i.(i7 
-;i.5l 
-4.05 
-3.57 


-3.59 
-3.37 

-3.7« 
-3.50 


-3.40 
-3.29 


—4.23 


9 
0 


10 


— 3.(;;i 

—3. 78 

—4.09 

1  —3.77 

-3.77 

—3.70 

—4.01 

—3.59 

7 

10 

8 

9 


7 
12 

8 
10 


11 

8 
8 
3 


s 
— 3.3fi, 
—3.35, 


— 3.3(J 

—3.27, 
—3.30, 

—3.29 

—3.21, 
—3.19; 


-3.20 


—3.84, 

—3.89, 

—3.8(1 

--.82, 
-3.(;3, 
—3  97, 
—4.10:, 
— 3.8l„ 

— 3.S7 

—3.81, 
—3.89, 
—4.02, 
—3.  ('.3; 


—3.87 

-3.71, 
— i-70, 
— 4.0(i, 
—3.(11, 


—3.77 

—3.(13, 
-—3.5(1, 
— 3.79.J 
—3.(10, 

—3.03 

—3.59, 
—3.30, 

—3.52 


-4.20, 


—  4.0 

—  (1.7 


—  0.53 

—  4.1 


-  .  5.0 

—  4.5 


-10.5 
-  8.9 


—  9.8 

—  0.3 

—  9.3 

—  12.5 


-  8.8 

-  8.4 

-  8.0 
-10.4 


-  8.0 

-  8.5 

-  7.9 
-10.2 


-  0.1 

-  7.P 

-10.2 


8.0 
8.8 


-13.4 


No.  of  I  Corrected  Loni;itudu.|  Latitude, 
obd. 


12 
7 
1 


C 
10 

8 
9 


8 
II 

4 
10 


11 

7 


—6.8 

-5.. 3. 
—5.3. 


[-45.8] 
—48.1 


[+n.2] 
4-11.0 


—5.3 

-4.4, 

-5.7, 

—5.1 


-9.. 5, 
-9.O., 

-9.0 

-  **-8, 

-10.3,., 

-  9.3; 
-11..5,, 


—9.0 

-T.8. 
-9.4„ 
—8.(1, 
-9.43 

—8.8 

_7.fi, 
-9..5„ 
-7.0. 

-9-2:, 

—8.0 

-S.l„ 
-8.9, 

—9.2, 


[_44.4]  [+11.1] 
40.9       -fll.4 


[_4;i.O]  [  +  10.7] 
—45.0       +11.2 


[— W.3]  [  +  11.5] 
50.5       +11.8 


[-54.0]  [+11.0] 
50.0       +11.5 


[-M.4]  [+11.0] 
—50.2       +12.1 


[-51.8]  [+11. .3] 
54.8       +11.7 


[-5fl.fi]  [  +  11.1] 
8.5        —52.8       -j-U.O 


-0.0, 

-8.8^ 

—8.9 


—12.5 


[—SO.O]  [+10.9] 
—51.5    I   +10.1 

[—(12.9]  [+11.5] 
—03.2       -j-12.0 


184 


T  H  K   O  U  IJ I  T  O  V   U  U  A  N  U  S. 


Mkan  Coukkutions 

TU  TIIK  Kl'IIKMt 

iiix  or  Uranus. 

(>bii«rv«(l  correct! 

—  Ciiiithiticil. 

Obs«rv«(l 

corrvvtiouH  in  It.  A. 

ma  ill  Uec. 

Cnrr.  to  (l«oceiitrio 

ObHerviitnry. 
K.  A.  of 

Mvau  dates. 



.... 

1 

Urauua. ) 

Moan. 

No.  of 

Ciirrecldd 

No.  of 

CoiTfcl«d 

Loiigltudv. 

Latitude. 

ulw. 

Ult'ttU. 

Muan. 

" 

obH. 

ui«nu. 

1887 

H 

H 

H 

'/ 

It 

Oreonwicli, 

Ang.     IS 

— 4.;!0 

10 

-4.33, 

—  13.4 

10 

-12..5,,, 

Cainhritljrt!, 

Aug.     IH 

— 4.0!l 

14 

-4.2S, 

—  12.(i 

11 

-13.4,, 

Eiliiihurgli, 

Aug.    22 

—4.40 

() 

—4.40, 

—  12.7 

(i 

-1-^.T. 

Paris, 

Aiij?.     14 

—4.34 

9 

-4.3,3, 

—  12.0 

10 

-11.4,„ 

Vii'iiiia, 

Aug.    2'> 

— 4.2'J 

4 

—4.33,, 

— 12.(! 

4 

—  11.(1, 

1 

[_(i2.9]  L  +  II.7JI 

[22"  Sfi-"] 

Aug     18 

—4.33^ 

—  12.5 

—(14.1 

+  12.2 

On'cnwifli, 

Si'pt.    17 

4.23 

14 

— 4.2(!,.i 

—  13.4 

14 

-12.5,, 

KoiiijfttlR'rj;, 

Sept.     11 

4.10 

H 

_4.13„ 

CuiMl)ri<l);(>, 

Sept.     17 

—4.  or, 

14 

-4,25„ 

—11.9 

15 

— 1-'  7 

Kdiiiljurgli, 

Sept.     10 

—4.311 

4 

—4.39, 

—  11.5 

3 

—  11.5, 

Piiris, 

Sept.     IS 

—4.20 

12 

-4.19„ 

—  12.3 

13 

—  11  7;., 

Vii'niin, 

Sept.    l;{ 

—4.12 

5 

_4.ir.. 

—  13.9 

6 

-12.9, 

[-(11,7] 

[  1  ll.fi] 

[22"  32"'] 

Sejit.    k; 

.... 

4.21 

.... 

—12.3 

—(12.3 

+  11.5 

Orccnwk'li, 

Oet.      k; 

—4.13 

11 

_4.1(V, 

—  12.9 

11 

-12.0,, 

Ciuiibridgc, 

Oel.      17 

— 4.00 

10 

-4.19. 

—  ll.(i 

11 

—  12.4,, 

Paris, 

Oet.       17 

— I.O;-) 

4 

—4.04, 

—  11.2 

4 

-10.(1. 

Vii'iino, 

Oet.       IS 

—4.41 

2 

-4.47, 

—  15.5 

a 

— 14.5i 

[-59.  S] 

[H1.4] 

[22"  2S'"1 

Oet.      17 

... 

—4^  1(1  " 

—12.0 

—(11.6 

-1  11.3 

Orcenwicli, 

Nov.       S 

—4.10 

5 

—4.13, 

—  12.4 

5 

—  11.5. 

Caiiiliridgp, 

Xov.       4 

— 3.1M1 

4 

— 4.ir,; 

—  12.3 

a 

-1.3.1:; 

Paris, 

Xcjv.       0 

—4.07 

3 

— 4.0(1' 

—11.3 

3 

-10.73 

Vii^iiim, 

Nov.       2 

—4.0!) 

2 

-'■'i 

—12.9 

2 

-11. 9j 

[—59.0] 

[-111.2] 

[22"  27"'] 

Nov.       (J 

—  1.11 

—11.7 

—(10.7 

-f  11.3 

Grponwicli, 

Dee.       2 

— 4.().'i 

2 

—  4. OS, 

—  14.0 

2 

-13.1, 

Cniiil)ri)i};e, 

Nov.     30 

— 3.S7 

(1 

— 4.0(1.. 

—  11.(1 

7 

—  12.4. 

I'orls, 

Dee.        H 

—4.03 

7 

—4.02; 

—  10.9 

7 

-10.3; 

Vifiiiia, 

Dec.        S 

—4.28 

2 

—  4.3l; 

—13.0 

2 

— 12.0( 

[_.<,7.7]  L+"OJ| 

[22"  28""] 

Dec.        5 
1838 

—4.05 

'   '    '    ■ 

— ll.(! 

—59.9 

+  11.1 

Oropnwicl), 

A  ug.    20 

—4.72 

11 

—4.70, 

—15.9 

n 

-1.5.1,, 

K(Mii(?sl)crjr, 

Aug.    2.') 

—4.  (-.5 

.5 

_4.(1.\ 

—  IS.C, 

5 

-17.(V, 

Ciuubridjtc, 

A  ug.     1 '.) 

— 4.07 

11 

—4.7  s., 

—  15.S 

12 

—  1(1.3,,, 

Kdinburgli, 

Aug.     2.T 

— 4.!)(1 

3 

_4.1I.V, 

—  15.0 

4 

-15.0. 

Paris, 

Aug.    20 

4.  SO 

9 

—4.79, 

— 1(1.(1 

9 

— 1(1.0„ 

It 

[-70. 9]  r4-ll.7l  1 

[22"  51"] 

Aug.     21 

... 

—4.7(1 

—  15.8 

^-71.4-' 

l_     1                    -J 

+  12.3 

Qrwnwifh, 

Sept.     12 

4.(12 

S 

-4.(1(1., 

—  1(1.2 

8 

-15.4, 

Koiii(»sl)iTg, 

Sept.     1)> 

— 4.tl.'-. 

u 

—  4.(17,,, 

—IS.  5 

14 

—17.5, 

raml)rid)?e, 

Sept.   k; 

4.(11 

11 

—4.72, 

—  14.9 

12 

-15.4,, 

Edinburgh, 

Sept.     15 

—4.74 

9 

-4.7;!, 

— 15.S 

1 

— I5.S. 

Paris, 

Sept.     12 

—4.71 

(i 

—4.70., 

— IC.O 

(! 

-15.4, 

Vii'iiiia, 

Sept.     11 

—4.71 

12 

-4.7.3, 

— IS.l 

12 

-17.1, 

Berlin, 

Sept.      t; 

—4.72 

7 

—4.74, 

—17.5 

1 

-1(1.5, 

[_70.4]'[+11.7] 

[22"  47°'] 

Sept.     1;') 

.... 

— 1.70 

.... 

—1(1.0 

—70.7       -\-\'i.i) 

III 


.,. 


THE   ORBIT   OP   URANUS. 


185 


., 


Mean  CoftRKoxioNs  to  the  Ei'iikmerib  or  U 

lANUS. 

— Continued. 

01)imrvnlnry, 
(U.  A.  »r 

Mean  (lat«a. 

Obaervi'd  oorreotlona  in  R.A 

OI>serTed  correotlona  In  Deo 

Corr.  to  Oeooentrio 

1 

1 

UraiiuH.  J 

Mean. 

No.  of 

CiirroeteJ 

No.  of 

Corrpoled 

Longitude 

batltnde. 

R 

uIm. 

liieau. 

Mean. 

obs. 

lueau. 

ff 

! 

1888 

8 

ff 

If 

,, 

Orodiwicli, 

Oct.      15 

— 4.fi7 

1 

-4.71, 

—  15.5 

7 

—14.7, 

CaiiiliridfTc, 

Oct.      17 

—4.49 

7 

— 4.fiO, 

—  15.3 

G 

—  15.M, 

Kdhilnii'^li, 

Oct.     k; 

— 4.5fi 

12 

—4.55:, 

—  15.0 

8 

—15.0, 

Piiri.s, 

Oct.     k; 

—4.(11! 

7 

— 4.fi5, 

—  15.9 

7 

—15.3, 

Viciitia, 

Oct.      14 

— 4.00 

9 

— 4.fis', 

—17.3 

9 

— 1(>.3, 

[22"  44'"] 

Ocl.^K! 



—  4.fi3 

—  10. 2 

[— CH.f,]   [f  11.5] 
— C9.7    1   +10.8 

Qreciiwicli, 

\(iv.     n 

—4.52 

fi 

— 4.5fi, 

— Ifi.O 

fi 

—  15  2, 

Ciuul)ri(l(fo, 

Nov.     1.') 

— 4.;i7 

10 

-4.48, 

—  15.1 

9 

—  15.fi, 

K<liiiliiii'<;li, 

Nciv.     k; 

—4.70 

s 

— 4.fi9, 

—  17.4 

2 

—  17.4, 

Vk'iina, 

Nov.       7 

4.93 

4 

—4.95, 

—  18.3 

4 

-17.3 

i 

■         1 

r    t\i\  01 

Till    m 

L — tiiL.ij  H  -t- 1  ■.•>  [  1 

[22"  42-"] 

Xov.      14 

-4.^58 

—15.8 

— «8.8 

+  10.9 

Oropiiwicli, 

Dec.       0 

4.fi0 

2 

— 4.fi4, 

—  15.9 

2 

—15.1, 

Ciiinliridp', 

l>cc.      15 

— 4.2fi 

5 

—4.37., 

—  15.1 

7 

-I5.fi 

Eilinhtirgli, 

iKc      17 

—4. IS 

3 

—4.17; 

—14.7 

1 

-14.7; 

Paris, 

Due.        5 

—4.40 

7 

4.39, 

—15.1 

7 

—  14.5, 

[— fi5.8] 
— «5.7 

[+11.1] 
+  10.4 

[22"  43"] 

l>cu.      10 

—4. 38 

—15.0 

1838 

Qri'diwieh, 

Aug.     22 

— 5,2S 

5 

—5.32, 

—21.3 

5 

— Sfi.fij 

Cutnbridgi', 

Aug.     24 

—5. 1 1 

S 

-5.22, 

—20.7 

7 

—21.0, 

Paris, 

A\ig.     23 

■■.  2  » 

3 

-5.1 9: 

—20.7 

3 

— 20.I3 

fidinbiirgli, 

Aug.     25 

—5  . ) 

2 

—5. 20, 

[-79.7] 
—79.7 

[+11.7] 
+  11.0 

[23"  fi"'] 

Aug.    23 

—5.22 

.... 

.  . . 

—20.7 

Grooiiwicli, 

Sept.     10 

—5.14 

12 

-5.18, 

—21.0 

12 

—20.3,, 

Kolliff.-iblTg, 

Sept.    12 

—5. 1 1 

12 

—5.13. 

—21.7 

12 

— 20.7„ 

BlTJitl, 

Sept.    10 

—5.10 

14 

—5.11. 

—21.1 

14 

—20.1, 

Cainliridgc, 

Sept.     17   '■ 

—5.12 

11 

-5.23: 

—20.0 

10 

— 20.3,„ 

Paris, 

Sept.     14 

—5.23 

10 

—5.22. 

—20.7 

10 

— 20.1,„ 

Ediiiliiirgli, 

Sept.     If,   i 

—5.10 

17 

-5.15; 

—20.5 

5 

—20.55 

ViiMiiia, 

Sept.     17   1 

—5.32 

9 

-5.33, 

—20.8 

9 

-19.8, 

[-V9.0] 

—78.8 

[+ ii.fi] 
+  11.0 

[23"  ;i"'] 

SeptrU 

—5.17 

—20.3 

Gn'ciiwicli, 

Oct.      12 

— 5.1H 

5 

—5.22, 

—20.2 

5 

—19.. 5. 

Caiiiliridjfo, 

Oct.      1.". 

—5.07 

S 

—5.18, 

—19.2 

5 

—19.5, 

Eiliidxirgli, 

Oct.      15 

—5,14 

10 

-5.13, 

—  19.9 

8 

—19.9, 

ViiMiiia, 

Oct.      10 

—5.17 

13 

—5.18, 

—20.5 

13 

—19.5, 

[_77.4]Y+11.5l| 

[22"  50'"] 

Oct.       14 

—5.18 

—19.6 

—78.7 

+  11.8 

Grccnwii'li, 

Nov.     12 

—4. 98 

fi 

—5.02, 

—20.2 

fi 

—19.5, 

Cnrnhridgo, 

Nov.     Ill 

— 4..S5 

fi 

— 4.9fi., 

—  19.1 

5 

—19.4, 

Paris, 

Nov.       !» 

—4.99 

2 

—4.98, 

—20.4 

2 

—19.8, 

Ediiiburgii, 

Nov.     1 1 

—4.90 

3 

—4.89, 

—18.3 

1 

—18.3, 

1 

r-75.3]  r+11.3]| 

[22"  57'"] 

Nov.     14 

.... 

—4.97 

—19  i    j  --75.7    1 

+  10.4 

Grcenwieli, 

Deo.       fi 

—4.84 

2 

—4.88, 

—19.0 

2 

—18.3, 

Cambridge, 

Dec.     15 

—4.  S3 

5 

—4.94., 

—19.1 

2 

—19.4, 

Paris, 

Dec.        3 

-4.90 

3 

—4.89, 

-17.8 

3 

-17.2: 

Ediiibursli, 

Dec.      2H 

— 4.9(! 

4 

—4.95; 

—  18.5 

3 

—  18.5, 

1 

'    [—73.9]  [  +  11.0] 

—18.2    j  -74.(1       +11.2 

[22"  57'"] 

Due.     12 

.... 

—4.92 

l'.]Cy 


TIIR   OIiniT   OF    URANUS. 


MkaN  C'llUllKl'I'lilNS  Til  TIIK  I'll'IIK.MKIllH  111'   UllANIIH.  —  O'lilli II  liril. 


Ubattrfntory 
[It.  A    of 
Urnniin.  ] 


Orccnwic'li, 
Ciiniltriil);)', 
Eiliiiliiir);li, 

[l>3''  20'"] 

OriMMiwich, 

Kdiii^slii'r)r, 

Cuiiiliriil)!0, 

Kiliiiliiir(;li, 

I'liris, 

Berlin, 

Viuiinii, 

[2;!''  IH"] 

(Ireonwicli, 

Ciiinliridtfc, 

Kiliiiliiii'gli, 

I'lirU, 

IK'rliii, 

Vii'iiiiiv, 

[23''  14""] 

Orcciiwicli, 

("niiiliridgo, 

Kdiiibui-jjcli, 

I'liris, 

ViiMino, 

[23"  12"] 

Ort'oii\vi4'li. 
Ciiinln'idjji', 
Kdiiiliurj;li, 
Vii'ima, 

[23"  12""] 


Groonwicli, 
I'liris, 

[23''  ST"-] 

Oi'oonwicli, 

KoiHgsbi'i'g, 

Berlin, 

Edinbiirgli, 

Paris, 

ViiMinn, 

[23"  33""] 


OWwrvoil  currHullniin  In  R.  A.    OIwitvhI  ciiiTi'ilidiiH  In  I>i'i'.      Ciirr   tii  (Iciii'i'iitrlo 


No.  of  I  Corri'uti'd    LiingltuilH     liStituile. 
oIjm.         iiioiiii. 


-23,!),, 

-2r).2, 

-23.7, 


[_S7.!.]  [  +  11.4] 
23. !l       — S«.8       -j-l''" 


-23.4. 
-24.1, 

-2..,\ 
-23.(1. 
-•>:,  3, 
-21.2, 


—23.8 

-23.0, 

-^3.7, 

-23.2,,, 

-22.7, 

— 2I..S 

-21.5, 

—23.3 

00  n 

--" « 
-24.4, 
-22.  H, 
-2.''). 5. 


[_87.0]  [+il.5] 
—87.5       -f-11.4 


—23.3 

-22.7, 
-22.5., 

—22.8 


2.S.1, 

2il.O, 
28  3 

2S.5,„ 

2-.U., 

2«.r,, 

2S.(), 

28.4,, 

29.3, 

[_85.8]  [  +  11.4] 
—8(1.3      +11.2 


[_S3.7]  [  +  11.3] 
—84.4       -4-10.3 


[—82.2]   [  +  11.0] 
—83.7       +10.4 


[— !)fi.!l] 


-2S.. 


[-nfi..5] 

— i)(!.3 


[  +  11.3] 
+  10.G 


[  +  11.4] 
+  l(».4 


M    * 


1 


TllK    OUUIT    Ol'    UUANUS. 


131 


(i 


1 


MkaN  ('<lUltECTU»N8  TO  THE  Kl'IIK.MKIllS  Of  U 

ii\svH.—Conlinui:il.                           1 

OlmervBtory. 
[11.  A.  of 

Obaisrvtiil  norrvctionn  in  R.  A. 

Ol)«ervcil  cnrwctloiis  lu  l)«o. 

Corr.  to  (iHoinntrlo 

Meau 

dnt«8. 

1 

"" 

UiaiiUH.  ] 

Mean. 

No.  of 

CnrrHcrltiil 

M«an. 

No.  nr    rorrei:U'il 

tiOii|;ltU(lH 

t.utttiide. 

8 

(ibrt. 

lUUIlll. 

H 

If 

1           " 

/» 

1841 

II 

Orcciiwicll, 

Oct. 

17 

—CM 

t 

— r..]:i, 

— 2S.2 

7 

'  —27.7, 

Bi'i-liii, 

Oct. 

20 

— .'■>.S7 

2 

—5,, ST, 

—  2S.2 

2 

7 

—27.2, 

Kdiiihurgli, 

Oft. 

la 

— (i.OO 

4 

-(1.04, 

—27.3 

4 

1 

—27.0. 

Piiri.s, 

Oct. 

21 

—11.10 

1 

— (l.O'.t, 

—27.2 

1 

-2(1.(1, 

V  it'll  mi, 

Oct. 

l(i 

—(1.14 

"J 

—(1.14, 

— 2S.S 

!) 

:  —27.8. 

[-94.0] 
—91.1 

[+11.2] 
+  10.5 

[2a''  2S'"] 

Oct. 

IS 

—(1.0(1 

...     —27.4 

Grconwicli, 

Nov. 

17 

— .'■..n.-> 

5 

— 5.09j 

— 2S.0 

5       —27.5 

BlTlill, 

Nov. 

111 

— (1.1;-) 

2 

-(1.15, 

— 2s.  ^ 

2       —27.4,' 

Etiiiiliiirgli, 

X..V. 

2 

'  — f).8!» 

a 

— 5.!ia, 

—2(1.5 

4      — 2r,.:t, 

Yiuiiim, 

Nov. 

U 

— f>.!KS 

4 

— 5.!)S, 

—30.0 

4       —29.0 

[2a''  2(1'"] 

Nov. 

la 



— (1.01 

—27.2 

[-92.0]  [-1  11.0]  1 
—93.3       -f  10.4    1 

Grooiiwicli, 

Dec. 

14 

—5.8(1 

H 

— 5.!»0, 

—27.4 

8 

—2(1.9. 

Bi'i'liii, 

]).c. 

l.i 

— ."i.dO 

a 

— 5.(10| 

—25.5 

■  3       —24.5, 

K(liiiiiiii';.'li, 

Dec. 

20 

— .^.(iH 

4 

—5.7:1, 

■ 

Puri.s, 

Dec. 

1,^ 

—5.7s 

4 

— 5.77j 

—2(1.7 

4 

—2(1.1. 

[2a"  2(;"'] 

Dee. 

k; 

— 5.7s 

^            4 

— 2U.¥~ 

[—90.5] 
—89.9 

[  +  10.7] 
+  9.8 

1842 

GrocnwU'li, 

Aiijr. 

lit 

— (1.(12 

5 

— fi.r,4, 

32. 1 

(i 

—31.7, 

Ciiinliriilcc, 

AUR. 

24 

— (i.4s 

2 

— fi.55, 

-32.7 

2 

-33.0, 

[2a"  ol'"] 

Ang. 

21 

—(1.(11 

't 

~32To' 

[—105.(1] 
—  103.0 

[+11.1] 
-t  10.0 

Orponwicli, 

Sept, 

12 

— fi.,W 

4 

—(1.57, 

—32.2 

4 

-31.8, 

KoniR.sbiT}?, 

Sept. 

is 

—(1.(1!) 

10 

—(1.110,  1 

-:!:;. 4 

10 

—32.4. 

Uorliii, 

Sept. 

11 

—(1.70 

3 

-(1.(1!.; 

— ;!1.7 

3 

—30.7, 

I'«ri.>!, 

iScpt. 

14 

— fi.(15 

5 

—(1.(14., 

—32. 1 

5 

—31.5, 

Caiiiliridpo, 

Sept. 

11 

— (1.(10 

1:5 

— (l.(17„ 

—31.9 

14 

—32.2,, 

K(iiiil)iir};li, 

Sept. 

2.! 

— fi.57 

4 

—(1.5!), 

—30.9 

4 

— 30.(lj 

Tulkowii, 

Sept. 

17 

-(1.(1  a 

c. 

— fi.(14, 

Vienna, 

Sept. 

1',) 

— c.sl 

2 

— (l.SI), 

—33.5 

2 

— 32.5t 

[2a"  47'"] 

Sept. 

iiV 

— (l.dG 

r._l 05.41  T-l-ll  oil 

—32.0 

—104.3 

L     1 J 

+  10.3 

Orc'oiiwieli, 

Oct. 

20 

— r,.(ir, 

12 

— fi.OS, 

—31.8 

12 

—31.4,, 

Borliii, 

Oct. 

2a 

— (1.54 

4 

-(1.5.3-; 

—30.7 

4 

—29.7, 

J'liris, 

Oct. 

17 

— (1.(14 

12 

-(i.(i:i, 

—31.5 

9 

— 30.!!,, 

Ciiinliridirp, 

Oct. 

17 

-(1.(10 

U 

— (l.fi7. 

—31.4 

10 

— 31.-,„ 

Kiliiilniri,'li, 

Oct. 

1.") 

—(1.57 

12 

— fi..'-)!!,' 

—31.1 

8 

—30.  .s. 

Vit'iiiiii, 

Oct. 

lU 

—(1.74 

4 

— (I.7:!; 

— 34.G 

5 

— ;):i.(i,  1 

I_r_l  03.01  r-4  11.011 

[2a''  44'"] 

Oct. 

17 

.... 

—(1.(14 

... 

—31.2 

—  103.7 

H  10.9 

(irrcnwicli, 

Nov. 

2.'! 

— (i.a(i 

7 

— fi.as,, 

—31.0 

— 30.r,, 

Berlin, 

Xov. 

H 

-(1.42 

2 

— (1.41) 

— :!0.2 

2 

— 2!t.2, 

Cniii1)ri(l!;e, 

Nov. 

17 

—(1.40 

7     ' 

—(1.47, 

—30.7 

7 

—31.0, 

iMliiilitirgli, 

Nov. 

111 

— (>.a!) 

=»  ; 

—(1.32, 

Vienna, 

Nov. 

20 

—(1.39 

a    ! 

— r,..ss. 

—30.0 

2 

— 29.0j 

' 

[_100.S][-f  lO.S] 
—  100.2      -flO.O 

i 

[23''  4.2'"] 

Nov. 

I'.l 

—(1.41 

.... 

1 

— 30.0 

18      May,  1873. 


wmmmm 


138 


THE   ORBIT   OP  URANUS. 


MbAN   CoUHrCTIONS  to  TUK  Kl'IIEMKRlS  »!»•   I'llAMS. <\llllinl(n!. 


[R.  A.  i.r 
Urauii.'t.J 


(llwfrvwl  I'orrt'Oliviis  in  R.  A. 


H«an  ilalini. 


!     Mi'iin. 


No.  of   CorrHi'tml 
(lbs.        uiuaii. 


Ort'fiiwifli, 

Uorliii, 

Piiris, 

Onml)ri(!!?o, 

K(linbui'(;li, 

Pulkown, 


1842 

Dfc.      K 

Dec. 

Doc 


Dff. 
Doc. 
Dec. 


[iJJ"  4r]        Dec.      K) 


EtiinbiirKh, 

Greenwich, 
I'rtriH, 

[0"  C"] 

Orcenwicli, 
I'liri.s, 
Kdinhiirfrli, 
Piilkiiwtk, 


!      1843 
I  Jiiri.         <) 

AiiR.  20 

Au-.  20" 

Sept.  17 

i  Sc|.t.  l,-)  1 

Sept,  l.S    ! 

I   Sept.  22 


» 
-C,.,-!! 
-G.Ki 

-(;.2;j 

-(',.22 

-(;.2.') 

-t;.2s 


— ii.:!2 

—7. II 
— 7.20 


-7.2.'J  1  ';) 

-7.17  :  II 

-7.2(i  1  7 

-7.1s  10 


O.-t. 

IS 

Oct. 

17 

Oct. 

17 

Oct. 

17 

\.IV. 

20 

Nov. 

11 

Xov. 

1.^ 

Nov. 

1.-. 

Nov. 

15 

[o"  ;i'"]      I  Sept.    ID 


Green'.vich, 
Paris, 

L(ii!|i)lirir||, 

[23"  5  It'"] 

Ori'emvicli, 
Koiii;.;slici'jf, 
I'ari.s, 
I'Mliiltiir};!), 

Oreciiwicii, 
I'Miiihurj!;,!, 
I'liris, 

r2:;''  ,-.(.'"] 

I  Jrcenwicl,, 
I'arir!, 

|;)ii   22"'l 

Orcciiwicii, 
Kiliiil)iir);li, 
KOiii)?slicrj?, 
I'liri.s, 

[0"   IH"-] 


1841 

•litii. 
.Ian.     I 

.Ian. 


Jan. 


Ansj.  I!» 

A  11,1?.  17 

Aug.  18 

Sept.  2.") 

Sept.  li» 

Sept.  17 

Se;>t.  10 


Sept.    17 


-7.10 

-7.07 

M.OJ 


-7.01 

-(1.S4 
-7.0;t 


10 
12 


-fi.7o      a 

-(!.C.7         (1 
-(i.72        2 


-7.70 


-7.7t 
-7.(i7 
-7.(!5 
-7.U3 


10 


II 
II 

10 
!0 


8 

-<i.i5, 
— (i.22, 

-(!.2',.. 
— (I.ISj 
— (!.2S, 

— (;.27 


7        — -O.".') 

I 

7     !  —7.0.1, 

a    i  —7.  lit, 
. ..     —7.10 


Olis»rv«d  rorrpvtioiin  in  Deo. 


Munn. 


■.■iO.2 
•.•il.4 
■HO,  It 
.•!0. 1 


-7.10, 
-7,10. 
—7.1!), 

-7,  IS, 

—7,17 

-7.0:!, 
-7  0(1, 

-(I.  as, 
—7.02 

— fi.iU,, 

-(!.s:{. 
—7 .  02_, 
— ti.Sd,] 

— 0,',)0 


— o.co, 

-0.71; 

— fi  a 
-7  (17, 

-7.  •;-.», 

—7. (IS 

— 7.(i.s, 
—7.(10, 
-7.(14, 

—7.02, 

— 7.(ia 


No.  of 
ob» 


-.T1.4 


-3-).  4 

-;!(;.  s 

-3(1.8 


-:!;">.  3 
-37.1 

■3,-i.l 


■34.3 
■34,4 
■3(1.3 
■3:..  7 


-34.1 

-.3.-..,-| 


-3!t,n 
-40.0 


-3<,).!» 
-40.3 
-40.7 
-31).  4 


0 
14 

5 


10 


10 

3 


II 
10 
10 
15 


Corrnittiil 
iiii'iin. 


-2!).S, 
-30.4] 
-3(1.3, 
-30.4. 


Corr.  lo  (ivovcnlrio 
Iiongituile.    Lulituil«, 


—30.2 

—31,0 

—34.7- 
-37.1, 

—35.7 

-35.0, 
-.3(1.2,. 
—3(1.4, 

— 35.S 


!4.i), 

id.  5, 

(4.7, 

55.2 

!3.')„ 

i3.4, 

'•'■  ',1 
!5.;{,. 

31.5 


-34. 
-34. 

-3!). 
), 


-31),  4, 
-30,2 


-30. 
-31). 
-30. 

-38.; 

-31). 


.  [—  n8.0][-|-l0..5] 
I  —  08.2  I    -f   D.G 

l~  !>'7.5]'[  f  1()..3J 

:—  1)8.2  ,    -[-   8.(i 

i 


[-II4.,3][-FI0.7] 

-111.1)    :      -f-      !).G 


[— II4.H[  f  lO.d] 
-1  12. ll'      \  1(1.0 


[-1 12.(1]  [f  10.. 5] 
-110.(1      -f   i).() 


[_n().ij[  1 10.4] 

—  108.(1      -f   0.5 


r— 1011,1  ir  f    !).!)] 
—  105.0       )-  8.2 

I 


[-122.3]  [h  10,2] 
—  121,2     -|-  0.(1 


■■•  [—122.0]  [-flO..(] 
—120.(1     +  D.a 


THE   ORBIT   OF   URAMJS. 


13i) 


Mean  Couiikctions  to  tiik  Kimikmkuis  t)F  Uh.  nvs. — Co'ilinuvd. 


OlxiorTatory. 
[K.  A.  of 
Uraiiua] 


Mean  datt'S 


I 


flrccnwii'li, 
K(IIiiImii')j:Ii, 

[0"  If)'"] 

Ori'i'invicli, 
l'Miiiliur);li, 

[0"   1-.""] 

Kdiiiliiirgli, 
I'uris, 

[0''   10"'] 


(Jrociiwirli, 
Kiliiiliiir^li, 

[O*-  11'"] 

(Jivcir.vicli, 

(Jri'iMiwirli, 

Kinii>;sl)>i'}f, 

I'aris, 

[o*  :j;r] 

(}i nwii'li, 

I'aris, 

[0"  iV] 

(Jri'i'iivvii'li, 
[0"  :;T'"] 

(In-ciiwicli, 
[0^   l-.V"] 

Ol'I'Cllwilll, 

I'uris, 
[tf  .Ml"] 

(Iri'i'iiwii'li, 

Konigsberj^, 

I'aris, 

[0"  41'.'"] 

(Jri'CMwirli, 
I'uris, 


1044 

Oct.  n 

Oct.  l:! 

Oct.  l.i 

N(iv.  2(; 

Nov.  1!» 

Nov.  2.! 

Doc.  1 S 

Iter,  i-i 


Obii«rv«iloorr«ctioii8  in  R.A. , Observed  corrections  in  Deo. 


Muau. 


-■.7t) 


,  No.  of   TorrtH'tcd 

Obl<.  UIKUU. 


l)c( 


20 


1845 

.laii.  I.'i 

.liiii.  14 

,laii.  14 

.Vug.  2r. 

Sept.  IS 

Sc|it.  ;!o 

Sept.  1 4 


Sept. 

Oct. 
Oct. 


j   Oct. 
!   Nov. 


I- 

20 


Dec.      1 1 


1846 

Sept.       S 
Si'pt.     12 

Sept.     II 


Oct. 
Oct. 
Oct. 

Oct. 

Nov. 
Nov. 


[0''  41'"]      j   Nov. 


s 

4 
14 


10 
III 

1  I 


.M 


-7.44 
-7.41 


-7.24 
-7.10 


^7. IS 
-7.il! 


-S.2.'> 

-S.21 

-S.lli 

-S.24 


-S.22 
-S.Ol 


-S.OC, 


10 


10 

6 


i   lU 

c 


—S.SO 

— S.7:! 


-S.7I 

-S.Cl 
_H.(1!» 


--s.r,2 

— S.4S 


H 
1!) 


7 

II 


8 

—7. 1!  4, 

^7.r.i, 
— 7.r.!) 

-7.:i.s, 
-7.:(5, 

— 7.;i7 

—7. IS, 
— 7.0'.>, 

—7.14 


.10, 

.OS. 


Iv. 


No  of  ('orrec;?  I 
Mean.     '  ub».  I     uioaii. 


—SI).  2         !t      — :!s.,s^ 
—40.,')  1    ;   —40.0, 

-a',).o 


—7. 

— s. 

s, 

— s. 
— s, 


(•S 

IT. 

1\ 

2:1, 

.17 

!•. 

00, 

.OS 

.OS 


—7.77 


— s. 
— s. 


.-;(..  7 


—.'!!).  n 


— ;!s.4 


-42.7 

-42.(1 
-4;!.  4 

-4;i.',» 


-42.2 
-4:i.7 


—4;!.:! 


-40.2 


_s.7.V,  _4C..2 
_s.72,„!  — 4(i.O 

— S.73  ;     .... 

_s.7o,  — n;.s 
_s  (;i„  -_4(  ■ 
— N.tis,    _4(;.', 

— S.ti;")    I      ... 


— s.,^s.^ 

— S.47, 
— H.M) 


-    (i.S 
-4(1.4 


9 


Corr.  to  Ouuoentrio 


Longitude. 


Latitude. 


L-I21.f.]  [  +  10.2] 
—  120.0    i    -f   11.4 


r-ns.2]  [-1-  '.).!)] 

— llii.d      +  H.!< 


'•   [-I1.V7]  [f  0.(1] 
_;i,s  7      -11,1.7        f   7.1 


10 

a 


10 
(j 


— as.o 

— as.o 
— 12.  a 

—42.1' 
—42.4, 

_4a.a; 
— 42.r. 

— ".s,„ 
— «a.  1,, 

—42.  a 

—42.0 


-114. 
-112. 

-i;ii. 

-120. 


n- 


[-lai  d]  [ 


-lai 

-12 


■lao, 

.I2s 

■  12S 

■  1211 


o.r,] 
7. a 

0.(1] 


0.0] 
0.1 


+ 

[+ 


« i 


s 

II 


..•lO.S     [-124. S]   [  I 


~4."..S 
—  111.:! 


—  I22.S 


ft..S] 
S.O 

0.(1] 
7.0 

-,).a] 

0.(1 


"'[-lao.oj  [  j    0.2] 
411.1   I  —las. 2      (    s.a 


7 
II 


-4(1.4, 

_4a.s. 
la    — 4(1.1  „ 


[-l;!0.7]  [  I    0.2] 
4,-,.s  I  — i;!7.l       I   s.a 


(1      ~M  4,  I  1 

0     — 4r>  s  '  I 

— LJL[-ia(i.M]  [f  0.0] 

_4(i.o  ]  — ia.">.i      I    7.(1 


140 


TlIK   ORBIT   OF    UKANUS. 


Mkan  Correct.ons  to  tub  Ei.,iKMLtt,8  o.  Viij,svs.-Continued. 


Ob8erT«torjr. 
[U.  A.  of 
Urauus.] 


Grccnwifli, 
Paris, 

[()'■  SO""] 

Green  wicli, 
I'liris, 

[0''  40'"] 

Oreenwieli, 
[■Jh   ,;,„-| 

Grcenwieli, 
Pur-', 

rjh  2'"] 

Green  Willi, 

Puris, 

[0''  ;.:'"] 

Greeti\vi(rli, 
Puris, 

[0"  a^-"! 

Greenwich, 
Pu-is, 

[()"  54"'] 

Greenwieli, 
Purls, 

[1"  !!»■"] 


Greenwieli, 
Paris, 


Groenwieli, 
Puris, 

[!''  ir] 

Greeiuvieli, 
Pnris, 

[I''  O'"! 

Greenwieli, 
Puris, 

[1-  <r'2 


M«aii  dates. 


Observe,!  oonectioim  ii,  R.A.  Observe.Uorreotiong  In  D™. 
Meau 


1846 

Dec.      15 
Dec.     k; 

Dee.     1<; 

1847 

Jan.      Hi 
Jan.      II 


<>aii. 
Sept. 


J  2 
3 


o<-t. 

Oct. 

I 
12 

Oct. 

10 

\<n-. 
Nov. 

Nov. 

« 

Doe. 
•Dee. 

i! 
12 

Doc. 

9 

1848 

Jan.      10 
Jan.      11 

Jan. 

11   1 

— 8.;}i 

—8.25 


— S.25 
— S.I4 


— tl.2I 


— fl.  25 

— !'.  2;j 


— !M0 
—II.  OS 


— s.no 

—8.84 


— S.8.1 

— 8.5:j 


Sept.      8  ,  —9.84 
.Sept.    22  ■  —9.82 

Sept.     17 


Get. 
Oct. 


[P  15'"]      '  Oet. 


19      _n.83 
li;      —9.79 

17 


No,,    i:)     _9.C8 
Nov.    i;i  I  —9.55 


Nov.     l;i  I     

i 
Dee.      14   '  —9.41 
Dee.        ',)      _<).4;t 


De. 


12 


1849 

Jnn.       n     — !)..^7 
Juii.      is   ,  — <).  11 

Jan. 


No.  of  Corrveleil 
obx. 


(i 
18 


9 
10 


T 
II 


11 


luuun. 


Mean. 


-8.27.,  I  — 4(!.0 
-8.24,     — 4(;.i 

—8.20 


-8.20, 
-8^12, 

—8. 15 

— 9.ir, 


—9.20, 

—9.21 

—9.0.-), 
—9.0(1, 

—9.01! 

—8.91, 

— 8.82J 

—8.85 


— 8.7.\ 

—9.79, 
—9.80,, 

—9.80 
—9.77. 


—45.4 


—49.4 


—49.0 
—49.2 


—49.2 
— 49.:i 


— (9.0 

—48.5 


—47.1 
—47.fi 


—5 1 . 1 
—51.5 


-!'.TT„:  -52.2 
—9.77 


— O.^.-i, 
— 9.5;t, 

—9.59 


— 52.:t 
—52.2 


— 9.:tfi,  I  _-51.3 
—9.41,     —51. a 

— 9.;i8 


— 9.:!T,     —.''1.2 
—9.09,      —50.0 


— 9.2:j 


Nn.of 
ob.i. 


10 
4 


Corr.  to  Oeocentrio 


Corrected 
uieuii. 


Longitude 


—»■>■«,„ 
—45.5,  ,. 

*  [— 1.34..5] 


—  i;J1.7 


[— 1.S0.9] 


lo 


— 45.0 

-42.7. 

_-44-«. 

—44.0   p— 129!g 

—49.0     [—148.4] 
—144.9 

-48,fi„ 

-'•':''iC-i48.o] 


—48.fi 

— 48  S 
-48.7,, 


-145.6 


[— 14fi.21 
-48.8    :  —142.(1 


—4S.fi, 
—47.9, 

—48.2 


-4r,.7., 

—47.0. 


—  4(1.9 

-50,7, 
—50,',),. 


[—143.0] 
—140.fi 


LatitDdf' 


[+8.9] 
+  0.5 


[+S.fi] 
+  7.4 

[+8.7] 
+  0.8 


[+8.7] 
+V.8 


[  +  «..'■.] 

+  7.2 


[+8.3] 
-f  (1.8 


[—139.4] 
— 13fi.2 


—50.8 

—51.8, 
-5I.«, 

—51.7 

—51.9, 
—51.0. 


[—150,4] 


[  +  8.1] 
+  0.0 


[  +  8.0] 


—  154.1        +7.5 


[-15fi.4] 
—  154.2 


"•»    ,  _151.8 


—  50.8j 

—50.7.1 

-50.7     [-'•'•>'>■  2] '[  +  7,7] 


[-fS.O] 
+  0.9 


[+8.0] 
+0.3 


-148.0' 


— 50..S, 
—49.4,  ' 

^      [-147.7]   [  +  7. .5] 
—50.1       -HO.a',    +0,(. 


+0.2 


THE   ORBIT   OP   URANUS. 


141 


.1 


Mean  Coukections  to  the  Ei>iiemeri8  ok  Uranus. — CoiUinued. 


Observatory. 
LU.  A.  of 
Uraiiua.  ] 


Oreciiwicli, 
[!''  ai'"] 

GiTomviili, 
I'ari.s, 

II''  yo'^j 

Ort'L'invifli, 
Paris, 

[1"  2(rl 

Qrcriuvicli, 
I'uri.-i, 

[1"  ar] 


Qrcrinvioli, 
J'lliis, 

[!'■  24'"] 

Qrccinvich, 
[I"  r.2'"] 

Orcciuvioli, 
Paris, 

[I"  47'"] 

Grcpriwifh, 
Pari.s, 

[!'■  4:r] 

Orccnwicli, 
Koiiifr--l»'r(r, 

[1"  41"- 


Orccnwicli, 
[I"  41'"] 

Orccinvicli, 

Koiiijtsltcrg, 

Puris, 

[2^  r-] 

KoiiiKshorg, 
Piiris, 

roll   2"] 


Mean  dates. 


Observed  corrections  in  R.  A.  Observed  correotions  ill  Deo. 


1849 

Scjit.     1(> 

Oct. 
Oct. 

2.S 
21 

Oct. 


Nov. 
.Nov. 


Mean. 


s 
—10.27 


—  10.24 
— 10.2(i 


K;  '  —10.12 
17     —10.22 

Nov.     17 


Dec. 
Dec. 


l:) 

lii 


Dec.     14 

1850 

Jan.       7 
•Ian.       .') 

Jan.        G 
Sept.      C 


Oct.  11 

Oct.  17 

Oct.  1(! 

\(.v.  r. 

Nov.  10 

Xov.  8 

Dec.  7 

Dec.  l!i 

Deo.  9 

1851 

Jan.  Ill 


Sept.  II 

Sept.  1!) 

Sept.  14 

Sept.  14 

Oct.  22 

Oct.  2:; 

Oct  23 


— o.n,') 


-  0.71 

— '.».77 


— io.,s;i 


—  10.02 
— I9.S7 


—10.70 
— 10.«0 


— 10..")0 
— 10..'')0 


—10.21 


—  11.42 

—  11.41 
—11.5!) 


— 11.7.^ 
—  11.4(i 


No.  of  Correcteil 
obi.        mean. 


Mean.       No  of  Corrected 
1  obs.       uiuttii. 


8 
-10.27 


—  10.24, 

—  10.24^ 

—10.24 

—10.12 

—  10.20,, 

—  10.17 

—  O.Oi"), 
--  O.MC. 


—  0.03 

—  0.71, 

—  0.75^ 

—  0.74 
—10.81! 

—  10.02. 
— lO.xri. 

—  10.87 

—  10.70. 

—  10..s7'j 

—  10.82 

— 10..'-.fi, 
-10.48, 

—10.54 
—10.21 

—11.42, 

—  11.4li 

—  11.40 

—11.75, 

—  11.44, 

—11.50 


— 5;i..'5 


— 5;{.9 
—54.8 


—54.4 
—54.1 


-5.3.1 
-54.4 


-53.0 
-53.0 


-53.0 


—55.3 
—50.3 


—.-.5.7 
-  -.(i.l 


— .5.-.0 

— 'vd.o 


—54.8 


-55.7 


-58.4 


3     —52.9 


Corr.  to  Oeoceutric 


Longitude.     Latitude. 


[-f  1(15.1]    [-f 7.3] 


-|-I(i0.8 


-fO.2 


[_ir,4.4]    [+7.4] 

— i.a.o      4-5.8 


[-102.5] 
—100.1 


[— iro.4] 

— 150.0 


r+T.3] 

-l-c.o 


[+T.1] 

-f5.7 


"•  i  [_i:,o.8]   [+0.0] 

52.5      —153.8    ;     +C.2 
i  i 

9     -53.5     [-171.7] '[  +  r,..5] 
.    ■      i  —108.1         +5.9 

—'•'•'     [_173..3]    [+0.4] 
.  .      —55.3      —100.0        +5.3 

fi  i  —55.3, 
4  '  —.'•.5.5, 

'[-172.4]    [+0.4] 
■55.4      -100.1    I     +5.0 

54.r., 

■  '_■__.'[— 100.0]    [  (  0.3] 
—54.0    ,  —105.0    .         5.2 


0  ' 
2 


—54.4    '[_103..'>]    [+0.0] 
-10O.4         4-3.6 


'    [-1S0.0]     [  +  5.7] 
—170.9    I     -f4  4 


[_ismj  [)r..8j 

—57.4    .  —170.1    1     4  4.3 


142 


TlIK   Oil  BIT   OF   URANUS. 


MkaN  ('0ltKEUTItlN8  TO  TlIK  Kl'IIKMKItIB  or  UUANUS.  —  Cimtiliuod. 


Observatory. 
IR.  A.  of 
Uraiiu:'  ) 


Orpcnwifli, 
Piiiin, 

Qreonwidi, 
Piiri.s, 

[1"  .is™] 

Orpciiwicli, 
Paris, 

[1"  54'"] 


Orocnwicli, 
I'liris, 

[I-  o-J"] 
Orci'Mwii'li, 

Qri'iMiwicli, 
I'liris, 

[•2"  18""] 

Orpcinvicli, 
I'liris, 

Ort'c'iiwicli, 

Koni)!sl)fr)^, 

I'urix, 

[2"  10"] 


Qrccnwii'li, 
I'uri.s, 


Mean  ilatei. 


Observvdcorrtiutiniia  ill  li.A.  <)lid«rvedaorractioD9  in  Uecv 
Mean. 


1851 

Nov.       3 

Nov.     a 


s 


Nuv. 


\i)v.     22 
.Nov.     U 

.Nov.     21 


I>cc-. 


22 
•J."-. 


Dfc.     24 
1852 

.lull.      11 
.lull.      14 

.lull.      12 
.Si'pt.     1 2 


Oct. 
(M. 


17 
2.! 


Oct.      Ill 


[2''  9"] 
Jreenwii'li, 

'liris, 
[2''  .lO""] 

Jrconwic'li, 
furis, 

[2"  J4'"] 


Nov. 
Nov. 


i:i 
1.^ 


Nov.     14 


Dec. 
Dec. 
Dec. 


lit 

Hi 
p.) 


Dec      lit 

1853 

.lull.      12 
.lull 


Juii. 

Sept. 

Sept. 


1.-. 
17 

in 


Sept.    17 


Oct. 
Oct. 


27 


Oct.      22 


—  ll.-.o 

—  II.4S 


—  II. 4.1 

— li.;i;» 


—II.  1.5 
—11.00 


— lo.sr. 

—  1(».S!» 


—  12.  Of, 


-12.2.! 
-12.0(1 


-11.72 
-11.07 
-11.77 


-11  42 
-11.40 


No.  of 
obn. 


-12.00        !, 
-12. OS       5 


-12,00       4 

-12. .'is  1     2 


-12.f.!l       7 
-12.00       r, 


CorrKuteil 
luxau. 


-IL.'iO 
-ll.40! 

-11.411 

-11.41! 
-ll.:!7, 

-11.42  : 


-ll.lo.l  --.57.2 
-lO.'.IS  I  —57.2 


Ntan. 


-.'■)7..'> 
-iiS.I 


-iis.;} 


—  11.(10 


-lo.s.v 


.  |(),S7,!  —50.5 
-10. so 


—12.00. 


—  12,2;!. 

—  12.04 


—  12.  IS 
— 12.00  : 

—  12.00'; 

— l2.o;i  ! 

-11.72, 

—  11.75, 

—  11.7:1 


—  11.42, 

—  11.44 

—  11.4! 

—  I2.0tt. 

—  12.50. 

3l2,fi5 

-12,00. 
— 12,04' 

—12,00 


-5(1.5 


-5-1.0 
—50.0 


-50.7 
-00.2 


— 5S,7 
— 50,:l 
—.50.0 


—,'.0.7 
—57.0 


— 5S.S 
—50.0 


No.  of  Correi'tail 
ub«.       luuaii. 


—57.1, 
-57. .5, 

—57.2 
— 57.;k 

-57.0, 

— 50.Sj 
—  .50.0, 

— 5(iT" 


— 50.S, 
—55.0, 

—50.4 
—50.1 

— 5S.2, 
—50.0, 

— .5S.4 

— 50.nj 
—50.0, 

—50.4 

-5s.:i, 
-5s.:t, 
— 5S.4., 

— 5s.;! 

—57.1 

—  57.1 

—50.3, 
—50.4, 

— 50.;j 


-.58.4, 
-5S.4,_ 

— 5S.4 


C'i>rr.  tu  (ieooeiitrio 


Luiigituile. 


Lulituil«. 


[-IS1.4] 
-17S.1 


[+•'■••7] 
4-4.0 


[_I70.0]    [  +  5.0] 
-177.0         -f;!.8 


[-175.7] 
-172.a 


[+5.:!] 

-i-3.8 


[_l;:i.l]    [,5  2] 
—  100.4    !     -|-;!.I 


[-1S7.4] 
-1H4.5 


[f4.0] 


[_ISO.O]    [  I  4.7] 

-1S7.2       f;i.i 


[-1SS.0]     [4-4.7] 
—  1S5.0         -|2.4 


[_IM4.S]     [f4.fi] 

lsi.:i       -1-2.8 


[_1S1.0]    [f4.4] 


—  170.8 


f2,0 


[_104.fi]     [fl.O] 


—  101,0 


-f  '.!- 


[-100,7]     [-1-3.8] 
-102,7         +1.7 


THE   ORBIT  OP   URANUS. 


148 


MKAN  CullllKCTIONti  TO  THE  Kl-lltlMKHIS  OF  UUANUS. Coutillllfd. 


Olisfrvatory. 
|K.  A.  of 
Uiaiius.  ] 


Qrcenwicli, 
1'uri.s, 

[2"  31'"] 

Paris, 
r^ii  St"'! 


Grociiwicli, 
I'liris, 

[2"  iir,-] 

Paris, 
[2''  2r."'] 

Orccnwicli, 

[2"  r,:r] 

(irccMiwich, 
Piiris, 

[2''  SO"] 

flrccnwicli, 

Paris, 

Haiiliuj^i), 

[2''  4S'"] 

(iri'cnwicli 

I'aris, 

Saiitiii;^(), 

[2"  -in 


(jrccnwiili, 
[2"   4-"J 

(ircctiwicli, 
Paris, 

Orcciiwirli, 

Paris, 

Santiajro, 

(Jrccnwicli, 

Paris, 

Santiago, 


Mean  dates. 


1853 

Nov.     12 
Nov.     l.") 


Observed  currectiona  ill  R.  A.    Observed  corructloiiH  iu  Deo. :      Curr.  to  (ieoceiitriu 


Mean.     |  No.  of 
oIm. 


Nov. 
Dec. 


14 
12 


1854 

Jan.      1.1 
.Ian.      20 


Jan. 
l''cl). 


15 


-12.  CI 
-12.G7 


—  12.28 


-12.02 
-II. s7 


-ll.H!) 


.K 


cpl.    21      _l.!.l-2 


Oct. 
Oct. 


2(!   '  _l:!,24 
2y     — 1;!,12 


Oct.      27 


Nov. 
Nov. 
Nov. 


12 
l(i 
17 


—  1:5.22 

—  I.!. 21 
— l;l.20 


Nov.     li)  j 

! 
Dec.    n 

l>cc.         It 
Dec.      1.-. 


-12.!).'! 
-l:i,H 
-I2.s.-i 


|)c< 


14 


1855 

.Ian.      is 


O.'t. 
Oct. 

Oct. 

Nov. 
Nov. 
Nov. 


l:! 
2."i 

IT 

II 
17 
It; 


Nov.    ir> 


Dec. 
Dec. 
Dec. 


Hi 
IS 


-l.i.Tl 
-I. •!.!»:! 


-I.'i.s.', 
-|:!.I12 
-l:!.s2 


-i:i.4s 

-l:!..'i2 
-Hi  .^.1 


Dee.      1,S 


Corrected  ;     Mean, 
lui'aii. 


14 

10 


7 
I 

10 


10 
3 


-12.(l.'),„ 
-12.(1.5 

-12.2C 


-12.02, 
-ll.sr., 

-ll.'Jti 
-11.87 


r>     —13.12 


-i:i.24, 

-l.t.lO, 

-i:i.22 

-i;?.22, 
-i:!.l!t, 
-l:!.203 

-l:!.20 

-12.03, 
-l:f.l2, 

-!2.S5, 

—  12.!iO 


-I2.5.V 


-i:!.7I„ 

—  l:i.".tl, 

—  l:!.7.s 

i 

'— l.'t.s.^, 

—  l.f.c.o, 
_I.!.S2. 

—  I.!. 7(1 


— .'■iS.fi 

— ."ifl.it 
— r)it.7 


~M.'.) 


■;'i.').S 


-.'■i7.C. 
-.'.i;.(; 
-ri7.2 


-.'■.0.4 
-50.2 


50.0 


-50.7 
-5S.1 


— 57.S 


50.0 


-5S.0 


Noor 

obn 


14 

8 


—  l:t.4S„,    —57. .'I 
'_l:!,50,     — 5i;.7 

i— i;!.5;i,  I  — 5S.1 
— l:i,50  i     .... 


Correctvd    Longituile.     Latitude, 
lueau. 


-50.0,, 
-58.(1. 

-58. 0 
-5S.4 

-50.2, 
-57.0, 

-58.0 
-57.G 


5S.2 

— 5S.7    '     I 
-5S.7   I     a 


-57.. 5, 
-57.0, 

-5  7.. '5 

-57.5, 
-5S.5, 
-.5H.I. 

;  —57.8 

— 5S.I, 

-5(;.7i 

-50.1, 

i   — 5S.0 


—5  (',.4 


4      — 


55 

0 

"t 

55 

"i 

55 

:i 

57 

1, 

5(; 

^. 

5(; 

.li. 

—5(1.7 


10 

— 5(i.8, 

:! 

—5(1.5, 

(> 

-57.5, 

—57.0 


[_inr,.o]   [-H!.t] 

—I ',12.0    1     -I^I.U 

[_i.,.T4]   [+:t.(i] 

187.0        -1-1.5 


[-188.1]     [  +  3..5] 
— 183.0    .     4  0.1 

[-184.5]'  [4  3..5] 
—182.1    I     -[0.7 

[-201.3]     [  +  3.0] 
—10(1.0        -i-1.7 


[-203.7]    [-f2.8] 
lOJI.O        -i  1.2 


[—203.0]    [-I2.T] 
—  100.1         -i  1.2 


[—200.1]    [4  2.(1] 
—  105.3    ;     40.7 


[-104. n]    [42-''] 
—100.1       41.1 


[—200.2]    [4I.X] 
—204.0    j     4  0.1 


r-200.4]  [41.7] 

—205.3        —0.2 


[-20(1.2]     [41.'1] 
,  —202.1    i    —0.1 


144 


TUE   ORBIT  OP  URANUS. 


Mean  Coruections  to  the  KpiiE.MGiiiii  or  Ubanus. — Con  Hinted. 


OlpucrTRtory. 
[II.  A.  of 
Uramid.] 


Grcciiwitll, 

Paris, 

[2"  OS""] 

Grcciiwieli, 
Paris, 

[3"  27'"] 

Qrci'iiwicli, 
Paris, 

[3"  22"°] 

Orecnwifli, 
Paris, 

[3"  K"-] 


(Irconwicli, 
Paris, 

[3"  la"] 

Grecnwii'l), 

[3"  40'"] 

Qrconwifh, 

Paris, 

KoiiigsliiTg, 

[3"  41'"] 

Orci'iiwicli, 

Paris, 

KOiiijj.sln'rg, 

[3"  3Cr] 

Orccinvich, 
I'uris, 

[3"  32™. 2] 

Orccnwirh, 
Paris, 

[3"  3r.O] 

Qreeiiwich, 
[4"  3"'.3] 

flrooiiwldi, 
Piiris, 


Ub.-iervt'd  uorreotiona  <n  R.A.  Obserrvd  corrections  111  Deo. 


Mean  dates. 


Mean. 


No.  of    Corrected 
olix.        iiicau. 


1856      I 
Jim.     22 
Fob.     10 

Jttii.     27 


Oct. 
Oft. 


10 
Id 


Oct.  k; 

Nov.  IS 

.Nov.  17 

Nov.  l.S 


Dec. 
Due. 


18 
19 


Deo.      19 

1857 

.Ian.      21 
Jan.      12 


Jan. 
Oct. 


19 

8 


Nov. 
Nov. 
Nov. 


II) 

19 


Nov.     1 1 

Pec.  1 1 
Di'c.  K! 
I»oc.        8 

Die.      13 

1858 

Jan.  17 
Jan.      Hi 

17 


Jan. 

Feb. 
Feb. 

Feb. 
Oct. 


II 
12 

II 
17 


Nov. 
Nov. 


s 
-12.97 
-12.87 


—  14.30 

—  14.38 


—14.23 
—14.32 


—13.92 
—14.11 


— 13..')9 
—13.87 


-14.08 


-14.80 
-14.83 
-14.77 


— 14..-..'i 
—14.48 
—  14.50 


—  14.13 

—  14.19 


—  1 3.  SI 

—  13.91 


0     1-12.97, 
2     1—12.85, 


10 
3 


13 

8 


10 
1 


'—12.94 

I 
-14.30. 

—  14.30, 

;— i4.3(r 

I 

—  14.23, 
-14.30. 

'—14.20 

1 

—  13.9,3, 
—14.09, 

—14.03 


-13.  CO,, 
;_13.85, 

'—13.00 
—14.09 


-14.81, 
-14.81, 

—  14.75, 

—14.80 

-14.. 50. 

—  14.40. 

—  14.54J 

—  14.51 


—  14.14, 

—  14.17, 

—14.15 


Mean. 


-50.1 
-54.8 


—53.8 


—54.8 
—54.0 


—55,9 
—55.0 


—5,3.4 
— 55.3 


—49.1 


—5 '.3 
—51.5 
—52.7 


—53.2 
—52.8 
—53.7 


— 52.8 
—52.8 


—  13.89, 

—13.83 
I 
-15.00       5      —15.00 


13.82,J  —52.1 
—52.7 


-40.4 


14     —15.10 
20  :  —15.19 


10 
9 


I 


[3"  58'».5]       Nov.     n 


_15.I7,„'  — 4S.2 
—  15.17, 


Noof  Corrected 
obs.  I     uicau. 


—.55.0, 
—54.0, 

—55.4 
—53.3 

—53.3 

I 
—54.3, 

\  —54.1 


Corr.  to  Oeoceutrio 


Longitude.  ,  Latitude. 


[-in9..5]:[+i.4] 


—194.7 


—0.5 


5  j  —55.4, 
0  i  —54.8. 


[-213.8]     [+0.7] 
—211.0    i    —1.4 


[-214.7]    [  +  0.7] 


-210.2 


—0.9 


[-211.0]     [  +  0.0] 


•207.1 


-1.2 


—55.1 

—52.9,, 

^~""''' '[-200.4]  [  +  0.5] 

—53.5    1  —202.3  —0.4 

I  i 

—48.0     [-210.8]  [—0.3] 

—213.0    .  -1.9 


—50., 8, 
—51.3, 
—51 


—51.2 


'[—218.9]     [—0.4] 
—215.5        —2.4 


5      —52.7, 

0  '  — 52.0„ 

1  !  —52.5; 

—52.0 
—52.3, 


13 
9 


10 
1 


[—217.1]    [—0.5] 
—212.2    :    —3.0 


[—211.8]    [—0.0] 


—52.0, 


—52.4    I  —207.5 


-2.8 


-51.0,„i 
so  r,    I 

"— 'i  [_2n7.n]   [—0.0] 

—51.7    I  —203.1    j    —3.1 
—45.9     [—220.0]    [—1.4] 


II 

10 


—47.7 
—47.0 


-210.2 


-4.0 


[—222.1]    [—1.5] 
47.7    i  —218.0    I    —3.8 


TUB   OIIBIT   OF   URANUS. 


145 


Mkan  C'oiiRKCTioNs  TO  TiiK  Ei'iiEMERis  OF  Uhanu8. — Continued. 


Ob8«rTatory. 
[it.  A.  «r 
Uranus.] 


Ciri'cnu'icli, 
I'uris, 

[3''  sa^.c] 


Orconwioli, 
Paris, 

[3*  50"".  0] 

(Jrccnwicb, 
Paris, 

[S*-  49"'.  3] 

Grccnnifh, 
[4"  20"". 8] 

Orconwifh, 
Paris, 

[4"  n-'.-i] 

Groonwirli, 
Paris, 

[4*'  12"".:.] 


Orccnwic'li, 
Paris, 

[4'  S-.O] 

Qreonwifli, 
Paris, 

[4"  V'.O] 

Ori'cnwii'li, 
[4"  40'".!)] 

Greenwich, 
Paris, 

[4*'  3.")'".  8] 

Oreonwii'h, 
Paris, 

[4'  31-.  C] 


Groonwicli, 

Paris, 


Observed  corrections  in  R.  A. 


Mean  dates. 


1858 

I)<c     Ifi 
Dec.     k; 

Doc.     10 

1859 

Jan.      19 
Jan.     10 

Jan.     lU 

IVl).     18 
Ftl).     1 1 

Feb.     14 
Oct.      2."( 


Mean.       No.  of   Corrwted 
I    ubs.         uieau. 


.\.>v.  17 

Nov.  IT 

Dec.  1(1 

l)<c.  U 

Dec.  14 

18C0 

Jan.  If) 

Jan.  15 


Jan.     IG 

Pel),     n 
Pel).        G 

Feb7"l5 
Oct.      13 


Nov.    li") 
Nov.    22 


Nov.    IS 

Dec.     12 
Dec.     12 


Dec.     12 

1861 

Jan.     13 
Jan.     13 


-l4.!ir, 

-14. 7S 


-14..^G 
-14.72 


-14. IS 

-14.37 


—15.43 


-15.5fi 
-l.-).i;l 


-15.43 
-15.4G 


-15.08 
-15.08 


-14.fi4 
-14.79 


—15.52 


-15.88 
-15.79 


-15.77 
-15.79 


-15.45 
-15.52 


[4''  2fi"».8]   I  Jan.     13 


-14.97, 
—14.87 


—14.57, 
—14.70, 

— 14.G1 

—14.19, 
-14.. 3,5: 


G      — 


14.28 
15.44 


12 


10 
6 


-15.57, 
|-1_5.S9, 

—  15.5S 

—15.44, 
—15.44, 

—  15.44 


-15.09, 
|-15.0fi, 

—15.08 

—14.05,. 
—14.77,; 

—14.07 
—15.52 


-15.89„ 

j— 15.85 

-15.78. 
!— 15.77, 

1-15.78 


Observed  corrections  In  Dec. 


Mean.      No.of  Corrected 
oits.        mean. 


It 

-19.2 
-49.5 


-48.9 
-4S.2 


-48.1 

-48.3 


-42.0 


-43.3 
-4.!.0 


-45.2 
-44.5 


-45.8 
-44.3 


—45.0 


—35.2 


-38  3 
-37.3 


15.49,, 
—15.47 


-41.5 
-38.5 


-40.5 
-40.3 


12 


— 48.7.J 
-49.3, 

—48.9 


-48. 4, 
-48.0, 

-48.3 
-47.0, 

-47.8 
-42.1 


[—220.7]    [—1.0] 
—215.1        —4.1 


—42.8 
—42.8, 

—42.8 


Corr.  to  Oeocentrio 


Latitude.    Longitude. 


[—210.1] 
—211.5 


[—210.9] 
—207.1 

[—223.2] 
-219.0 


[-1..5] 
—3.1 


[-1.0] 
—3.4 

[-2.2] 
—5.5 


"  [-224.7]    [—2.2] 


—44.7, 
—44^3, 

—44.5 


■221.9 


[—223.9] 


20.7        -4.9 


10 

3 


— 45..3, 

—44.8 
—44.5 

[ 
—44.5 

—34.7 


[- 


—4.0 


[-2.3] 


219.0]    [- 
210.1 


-37.8., 
-37.1, 


—37.0 

-41.0, 

—38.3., 

—40.0 


-213.9]    [- 
-210.4 

-223.0]  !  [- 
-218.3 


[- 


220.0] 
223.5 


-220.2] 
-223.2 


-40,1, 

-■•"•*'■   [-222.4] 
—40.1    1—219.3 


-2.3] 
-4.5 


-2.3] 
-5.0 

-2.9] 
-5.4 


[- 


3.1] 
5.8 


[-: 


3.2] 
;  8 


[- 


3.2] 
5.8 


19      May,  1873. 


116 


THE   ORBIT   OF  UK  AN  US. 


Mkan  Cohhbctiuns  to  tub  EPUEMERI8  or  UiiANUB. — CoiUinuvd. 


ObsprTRlory. 
[R.  A.  of 
Uraiiun.  ] 


flrcenwifh, 
I'arig, 

[4"  21-.0] 

Orcenwicli, 
[4"  6G"'.4] 

Orrcnwii'h, 

Paris, 

Washington, 

[4'  50".8] 


Greenwich, 

Paris, 

[4"  44°'.0] 

Greenwich, 

Washington, 

Paris, 

[4"  43M] 
Oreonwich, 

[5"  ic.a] 

Greenwich, 

Paris, 

Leyden, 

[5"  10".  7] 


Ornonwich, 
Paris, 

Wasliington, 
Lcydcii, 

[5"  4"'.fi] 

Greenwich, 
Paris, 

Wasliington, 
Leyilcn, 

[5*  2". 2] 

Greenwich, 
Paris, 

[5''  2"".  21 


OliaerTwl  correotions  in  R.  A.  Olwerved  corrHctions  in  Drv. 


Mvan  (latea. 


Uhru. 


No.  of  CnrrMted 
I    ob».   I     nioan. 


1861 

Feb.     10 
Feb.       5 


Feb. 
Nov. 


8 
10 


■  1.5.12  I     2 
■15.23  1     1 


— l.'i.!)!) 


Dee.  a 

Dec.  10 

Dec.  18 

Dec.  12 

1862 

Jan.  19 

Jan.  20 


Jan.  19 

Feb.  22 

Feb.  19 

Feb.  15 


Feb. 

XoT. 


19 
9 


Dec.  12 

Dec.  1 1 

Dec.  8 

Dec.  19 

1863 

Jan.  1 7 

Jan.  20 

Jan.  10 

Jan.  16 


Jan.  17 

Feb.  15 

Feb.  15 

Fel).  9 

Feb.  15 

Feb.  15 


-1(1.01 
-Kl.04 
-Hi.  10 


-15.07 
-15.72 


-15.  IS 
-15. :U 
-15.47 


— lO.l.J 


-10.27 
-10.27 
■10.10 


-15.90 
-15.94 
-15.92 
-10.03 


Afar. 
■Mar. 

Mar." 


4 
11 

fi 


10 
2 
4 

0 


1-15.1:5, 
—  15.20, 

—15.15 

!— 10.00, 

1—10.02. 
|_10.01„ 
.10.07. 

10.03 


—15.07, 
-15.09, 

—15.08 

— 15.1.<«, 
■—15.31, 
1—15.44, 


No.of  Correotoii 
M««n.       oIm.  I     mean. 


Corr.  to  OeocDDtrlu 


-40.7 


—15.30 


15.55 
15.47 
15.04 
15.57 

JO 

17 

2 

C 

15.. 10 
15.,30 

2 
4 

— 

—10.13 


—10.27, 

—  I0.24j 

-10.13, 

-10.23 


-15.90,, 
-15.91, 
-15.92. 
-10.00, 

-15.90 

-15.55,. 
-15.44,, 
-15.04, 
-15.54, 

-15.50 

-1.5.30, 
-15.27, 

-15.30 


-35. 1 
-34.1 


-•J  i .  :s 
-20.5 
-•>■.  '2 


— 2S.3 
—27.4 


—27.9 


-29.2 

-2.S.0 

-28.5 


-29.0 
-28.0 


-40.3 


Iiongitnile. 


Latitude. 


.    ^  —40.3 
—30.9        8  '  —30.5 


—33.4    i     4   '  —33.0, 
—32.8      13  !  —32.0,, 


-34.3         7 
-34.0    i     6 


—24.5 


—32.7 

-34. 7„ 
—33.9, 

—34.4 
-33.9, 
—  34.4, 


[—217.9]    [—3.1] 
—215.0    I    —0.0 

[—220.2]    [—4.0] 


— ?23.4 


—0.1 


—34.1 
—24.1 

-20.8, 
-20.  .3, 
25.2 

—20.3 


-27.9,,, 
-27.2, 

-27.9 


[-227.3]    [-4.0] 
—224.4        —0.4 


[—223.2]    [—4.0] 
—220.2        —0.3 


[-217.5]    [-4.0] 
—215.1        —0.0 

[-220.2]    [—4.7] 


—223.0 


—7.1 


[—227.0]    [-4.7] 
—225.4        —7.2 


I'LL  [—224.3] 


—27.9 

—28.4, 

— 2.S.,5, 
— 2S.0 


'  [—219.3] 
— 21G.0 


-28.0, 

"^':*''_  [-215.9] 
-28.2      —213.3 


[-4.8] 
-0.8 


[-4.8] 
—7.2 


[-4.8] 
—7.0 


THE   ORBIT   OF   URANUS. 


141 


Mean  Cuiiuections  tu  tiik  Kimiemkuih  ur  Uhanub. — Continued. 


ObRerT«tory. 
[K.  A.  Ill 
Urauui..] 


Paris, 
Lcydcii, 

[5"  34""] 

Greenwich, 
Leyden, 

[5"  29~J 


Washington, 
Leyileii, 

[5"  as"] 

\Vii<liiiif;ton, 
[f)''  21'°] 

Orccnwifh, 
[5"  21"] 

Pnris, 

Waxliiiigton, 

L(ydeii, 

(5"  4!)"] 


Oreciiwiuh, 
Parin, 

Wasliiiigton, 
Leydt'ii, 

[5"  45™] 

Wusliingtoii, 
Lcydcn, 

[5>'  41-] 

Oreciiwlch, 
Wusliingtoii, 

[fl"  17"] 

Orcenwicli, 

I/cydcii, 

Wttsliington, 

[0"  ll"] 


Mean  <lal«g. 


1863 

Nov.     2!) 
Nov.     14 


Nov.     21 


Di-c. 
Deo. 


I!) 
14 


Dec.      18 

1864 

Juii.      15 
Juii.      10 

Jan.      14 
I'eh.      15 


May       :{ 


Dec.  20 
Dec.  10 
Dec.      18 

Dec.     n 

1865 

Jan.       5 


Jan. 

15 

Jan. 

17 

Jan. 

10 

Jan.  15 

Feb.  14 

Feb.  13 

Feb.  14 


Oct. 
Oct. 


10 

18 


Oct.  17 

Dec.  0 

Dee.  7 

Dec.  IS 


Dec.     14 


Obaervad  corrautious  In  R.A.  ObBervvd  oorreotiODS  in  Duo. 


Mean. 


S 
-1<!..'J3 
-l(i.23 


-10.33 
-10.51 


-10.01 
-10.07 


-15.74 


—  15.57 


-10.32 
-1(;.22 
-lO.l'.l 


-10.2.S 
-10.22 

-It;.  11 

-10.03 


-15.77 
-15.85 


-15.50 
-15.59 


-10.13 
-10.20 
-10.17 


No.  of   Corrected 
obn.         lueau. 


B 
—10.30, 
—10.20, 

—10.24 

—10.32, 
—10.48, 

—10.30 


—10.01,, 
—10.04, 

—10.02 
—15.74 

—15.57 


—10.29, 
—10.22, 
—10.10, 

—10.22 


-10.28, 
—10.19, 
—10.  II, 

—  10.00, 

—10.13 

—15.77, 

—  15.82, 

—15.78 

—15.59, 

—  15.59. 

—15.59 

—10.13, 
—10.23, 
—10.17. 

—10.17 


M«an. 


-18.4 
-18.1 


-20.0 
-17.1 


—20.0 


-21.5 


22.2 


-13.2 
-11.4 
-12.4 


-13.7 
-15.1 
-14.4 
-14.7 


-15.4 
-15.0 


—  2.4 


4.8 
3.8 


No.of  Corrected 
obd.       lueau. 


II 
4 


-18.2. 
-18.1, 


Corr.  to  Oeoaenlric 


Lougitade.    Latitude. 


-18.2 


[—225.9] 
—224.1 


-19.0, 

'^''•''   [—220.8] 
-19.0      —225.9 


-20.0 


—21.8, 

—12.8, 
-12.4, 


—12.0 


-13.. 3, 
-14.9, 


[—224.0] 
-221.4 

[—219.0] 
—217.8 

[-210.1] 
—215.0 


[—225.1] 
—223. 1 


"'111,  [—223.4] 
-14.3       —220.0 


-15.0 


[—218.9] 
—217.4 


[—215.3] 
—214.0 


[-5.G] 

—8.3 


[-5.7] 
—7.0 


[-5.8] 

—0.7 

[-5.7] 


[-5.0] 
—7.3 


[-0.5] 

—8.4 


[-0.5] 

—8.8 


[-0.4] 
—8.0 


—8.1 


.-  2.0, 

—  2.0 

-  4.8, 

-  4.0. 

[—222.2]'  [—7. .3] 

—  4.7      —222.0        —8.8 


148 


THE   ORBIT   OP   URANUS. 


i 


Mean  CuKKKCTioNg  to  tug  ICi'iikmkhis  ur  Uhani'b. — Cotiliiiucd. 


OlmorTatorv. 
(R.  A.  or 
Unuaa.] 


Qrccnwicli, 
t'arix, 

WaHliington, 
Li-yilcn, 

[fl"  5"'] 

Oreenwioh, 

Washington, 

Luyduii, 

[C  0""] 

Greenwich, 

\Vti.shinglun, 

Leyilt'ii, 

£«"  O"] 

Washington, 
[li"  aT'"] 

Qrccnwich, 
VVusliington, 

Qrccnwich, 

Washington, 

Leyilen, 

[( "  31""] 


Greenwich, 

Washington, 

Lcydon, 

[-(jh  24m-] 

Paris, 

Wasliinglon, 

Lcydcn, 


U«an  dates. 


1866 

Jan.  10 
Jan.  1>2 
Jan.  17 
Jan.     17 


Jan.  14 

Feb.  14 

Feb.  14 

Feb.  14 

Fub.~14 

Mar.  12 

.Mar.  9 


Mar. 

3 

Mar. 
Oct. 

9 
1.3 

Nov. 
Nov. 

9 
15 

Nov.    14 


Dec. 
Dec. 
Dec. 


11 
12 
10 


Dec.     11 

1867 

Jan.  21 
Jan.  21 
Jan.       9 


Jan.  19 

Feb.  l.J 

Feb.  15 

Feb.  18 


[C  20™]     I   Feb.     15 


Greenwich,    ' 
Wa.shington, , 


Mar. 
Mar. 


[C  19""]     I  Mar.      6 


Greenwicii, 
Washington, 

[fl"  51"] 


Dec. 
Dec. 


11 

18 


Dec.     13 


Ubavrvvd  vorreutloni  in  R.A 
Mean 


R 

-1(1.20 
-lfi.04 
-IC.OO 
-lfl.02 


-15.81 
-15.81 
-15.75 


-15.47 
-15.49 
-15.59 


— 15.3fi 


-15.fi5 
-15.72 


-10.01 
-Ifi.OO 
-15.87 


-15.89 
-15.93 
-15.82 


-15.72 
-15. (i5 
-15.54 


-15.45 
-15.32 


-15.IJ1 
-15.70 


No.  of 
obi. 


10 
6 

8 


2 

10 


10 
7 
2 


5 

12 

4 


Correoted 
u«au. 


Ubucirved  uorreotions  in  Dec. 


Mean. 


— ir..20, 

— Ifl.OI, 
— Iti.Ot), 
— 15.99, 

—16.09 

-1.5.81,. 
—15.81. 
—15.72. 

^5.79^ 

—15.47, 
—15.49, 
— 15.5(;, 

—15.50 


— 15.3« 


7.2 

n.3 
7.« 
7.1 


9.1 

8.4 
8.fl 


8.1 
9.7 
9.0 


+  4.4 


—15.65,  -j-  3.5 

—15.72,,  4-  3.8 

—15.71 

— Ifi.Ol,,  -t-  2.S 

—Ifi.OO,  -t-  2.0 

—  15.84,  -f   l.« 

—15.99 


—15.89, 
— 15.9;{., 
-15.79, 

—15.90 

— 15.fi9, 

—  15.fi,5, 

—  15.51, 


-15.04 

—15.4.5, 
—15.32, 

—15.40 

— 15.fil, 
—15.70. 

— 15.C4 


1.4 
1.2 

0.3 


2.4 
2.0 
0.5 


2.9 
2.5 


8.5 
6.9 


No.  of  Corrected 
uba.       mean. 


11 

8 
8 


10 


9  —  fi.S, 

1  j  —  6.1. 

9  1  —  6. .5. 

6  —  7.1, 

—  0.6 


Corr.  to  Oeocentrlo 
Utitude. 


Longitude. 


8.7, 
7.3, 
8.6, 


—  8.2 

—  T.7, 

—  8.0, 

—  9.0, 

—  8.5 
+  5.5 


+  3.9, 
+  4.9, 


-I-  4.0 


10      -f  3.2. 

7      -f  3.1, 

i  +  l.fi, 

-\-  2.9 


5 
12 


4 

10 
3 


1.0. 
0.1. 
0.3, 


—  0.4 

—  2.0, 

—  0.9, 

—  0..5, 


1.1 

2.5, 
1.4. 


—  2.1 

+  8.9. 
+  8.0,^ 

+  8.5 


[—221.6]  [. 
—221.0 


7..1] 
9.0 


[— 217.3]'[. 
—210.9 


-  7-3] 

—  8.2 


[— 212.9]'[- 
— 212.9 


[-209.0] 
-211.2 


7.1] 
8.5 


[- 


7..5] 
7.9 


[-21.5.1]  [ 
—215.9 


-218.2]  [- 
—219.7 


7.8] 
8.3 


8.0] 
9.7 


[-217.8]  [_  8.0] 
218.3      —  9.3 


.214..5]  [—  7.9] 
—214.7      —  8.0 


[-211.1] 
—211.3 


[—213.1] 
—215.3 


[-  7.8] 
—  9.0 


C-  «■«] 
—  10.4 


THE   ORBIT   OP    URANUS. 


140 


Mean  Cobiikctions  to  the  Epiiemkuis  or  Ukanub. — Continued. 


ObiierTatory. 
LK.  A.  of 
Unuua.] 


Qrccrnvicli, 
[(;"  45'°] 

Qrpcnwk'li, 
Lvydcii, 
VVusliiiigtoii, 
Piii'is, 

[•;''  41'"] 

LcydcM, 
Wttsliirigton, 

[(!'■  3!)'"] 

Wiisliiiipton, 
[7"  l(i"'] 

Wiisliiiijrton, 
[7"  Hi"'] 

flrccnwicli, 
Wusliiiigtiin, 

[7''  lO""] 


Orpcnwirli, 
Wusliiiigton, 

[7''  T)"'] 

flrconwifli, 

Wiisliiiigtuii, 
Paris, 

[7"  l"-] 

flri'ciiwich, 
Wttsliiiijjfton, 

[(i""  59"°] 

Oreenwicti, 
[7''  Si™] 


Qrconwicli, 
[7"  25'"] 

Oroeiiwic'h, 
Washington, 

[T"  21-] 


Mean  dstea. 


Obiertttd  oorreotloua  in  R.A. 


Mean. 


1868 

Jan.     13 


Fcl>.  15 

I'VI).  5 

Fub.  13 

Ftb.  14 

Mur.  1 1 

Mill-.  19 


Mar. 
Oft. 

15 
18 

Nov. 

7 

Doc. 
Dec. 

25 
12 

Dec.     18 

1869 

Jan.      19 
Jan.      18 


Jan.     18 


Ffl). 
Fi'l). 
Fcl). 


14 
14 

10 


Feb.      13 


Mar. 
Mar. 

3 

7 

Mar. 
Deo. 

5 

8 

1870 

Jan.      19 

Feb. 
Feb. 

Ifi 
1(! 

Feb.      IG 


— 15.C7 


-15.43 
-15.43 
-15.50 
-15.41 


-15.02 
-14.91 


-14.57 
-14.83 


-15.24 
-15.1(i 


-15.28 
-15.31 


-14.81 

-14.87 


-14.50 


—14.73 


-14.58 
-14.49 


No.  of 
obit. 


-15.04  •  10 
-15.13  13 
-15.09      10 


10 


Corrected 
luuau. 


8 

-I5.r.74 


—15.4.3. 

—  15.40, 
—15.50. 
—15.38. 

—15^44 

—  14.99, 
—14.91, 

—14.95 
—14.67 


—14.83 


—15.24. 
— la.Ki, 

—15.20 


-15.28, 
-15.31. 


Ubaerred  oorrootiona  in  Deo. 


Mean. 


-15.31 


-f  0.9 


+  4.8 
+  5.9 
-I-  4.9 


-t-  4.0 
+  3.1 


-fl2.3 


-j-lfi.2 


-fl5.1 
+  14.0 


-1-12.0 
-1-12.9 


-15.04,,  -1-11.1 

-15.13„i  -1-11.3 

-15.0(1,  -1-11.7 

-Is. 08 


-14.81, 
-14.87, 


-1-10.8 
-fll.4 


-14.84 

-14.50,     -1-21.1 


—14.72. 


-14.57. 
-14.49, 

-14.55 


-1-19.0 


-fl7.2 
-flO.7 


No.of 
oba. 


Corrected 
uieau. 


10 
13 
10 


Corr.  to  Quocentria 


Loiigltade. '  Latitude. 


-f  6.7.  [-213.4]  [—  8.7] 
-215.5      —10.1 

+  4.0. 

+  CO, 

[—210.3]  [—  8.0] 


-f  5.3 

-f  4.0, 
+  4.5, 


—212.3 


-f-  4.0 
-1-13.3 

-f  17.3 


+  14.9. 
+  15.0. 

+  15.2 


+  12.4. 
+  13.4. 

+  13.3 

+  10.9,, 
+  11.7,. 

+  11. t 

+  10.6, 
+  11.9^ 

+  11.3 
+20.9 


10      +18.8 


+  17.0 
+  17.1 


—  9.7 


-205.0]  [—  8.6] 
-205  5    1  —  9.2 

198.3]  [_  8.7] 


—201.5 


—12.9 

I-  0.1] 
—205.0      —  9.3 


[—202.0] 


C-: 


200.5]  [_  9.2] 
210.3      —10.0 


[— 207.1] '[—  9.3] 


—211.4 


-10.3 


[—204.7]  [—  9.2] 


-208.1 


-10.2 


[—201.5]  [_  9.2] 
—204.8    j  _  9.3 

[— 197.1]  i[—  9.8] 
—202.1       —10.3 


[—199.3]  [_  9.9] 
—204.5      —10.8 


[—197.2]  [_  9.9] 
+  17.0      —201.9      —10.9 


150 


THK   ORBIT   OF   U  II  A  N  U  S. 


Mkan  CoBBECTioNH  TO  TUB  Ki>iiKMERi8  Of  XJRASva.— Continued. 


Oh»vtJtlotf. 
[K   A.  vf 

VtHUMi.] 


Orm^nwieli, 
Wa.sliiii)(ton, 

[7"  18"] 


Qropiiwicli, 
[7"  ■17"'] 

flreonwicli, 
[7"  41'"] 

Orpcnwit'h, 
[7"  3S"'] 

Orponwicli, 
[S"  11'"] 


Greenwich, 
Wusliiiijjtoii, 

[8"  (;■".  1] 

flrcenwioli, 
Washington, 

[8"  l-.O] 

(JrocMwich, 
VVusliington, 

[7''  57".9] 

Washington, 
Uruunwioh, 

[7"  57'". 3] 


M«SD  >U.ea. 


Obierruti  oorractiona  hi  R.A. 


M«au. 


1870 

Mar.     I  a 
Alar.     1 1 

Mar.    12 

1871 

Jan.       9 


Feb.  15 
Mar.  14 
Dec.     21 


1872 

Jan.       A 
Jan.     21 


Jan.  18 

Fib.  15 

Feb.  21 

Feb.  17 

Mar.  15 

Mar.  15 

Mar~15 

April  8 

April  8 

April  8 


8 

— 14.-->!( 
—  14. HO 


— IH.fl!) 
—13.92 
—13.82 
— 13.1G 


-13.32 
-13.31 


-13.23 
-13.24 


-13.01 
-13.07 


-12.79 
-12.77 


No.  of   Correi'teil 
uba.  I     luoan. 


0I>   irred  correotioD*  tn  D«a 


M«an. 


14 

3 


14 
10 


—  I4.2S, 

— 14.3(», 

^14.29 
—13.98 
—13.91 
—13.81 
—13.15 


—13.31 
—13.31 

—13.31 

—13.22 
—13.24 


—  13.23 

—13.03. 
—13.07, 

— 13.05^ 

—  12.79, 
— I2.7li, 

—12.78 


-flfi.9 
-1-15.9 


-f24.8 
-I-2I.8 
-f20.7 
-f  29.2 


-f2.S.3 

-f.7.8 


-f2P..2 

-l-2(;.4 


-f-25.4 
-1-24.5 


-f24.9 
-f25.0 


No  of  Corrauted 
ub<.       lueitu. 


14 


+  lf..O 
-|-l(i.3 


Corr.  to  Oeocvutiio 


Longitude. 


Utitod*. 


14 

8 


-f  1(1.2 
-f24.fl 
-f21.6 
-1-20.5 
-1-29.0 


-f2S.l. 
+  28.2, 

-f28.1 

-l-2r..o, 

+  2«.H, 


[-103.f.]  [-  9.7] 


—198.1 


—10.1 


[—190.0]  [-10.2] 
—  19(!.l    j  —10.7 

[—188.8]  [—10.1] 
-194.4    I  —11.5 

[-184.9]  [—10.0] 
—192.8    I  —11.5 

[-177.9]  [-10.4] 


-18(5.5 


-11.4 


-f2(i.3 


[—179.8]  [—IO.fi] 
188.3      —11.3 


[_179.3]  [—10.5] 
-180.0      —11.3 


-f2o.2. 

_-\--t^i_  [—170.4]  [-10.4] 
-f25.1      —183.7      —11.1 


-f25.3, 

-f-24.8, 

-1-25.2 


[—172.4]  [—10.2] 
1  —180.2    I  —10.8 


1 


TUB  Oil  HIT   OF   URANUS. 


Iftl 


1 


COHIIECTIONM  TO  HE  Al'I'MKI)  TO  TIIK  PolilTKlNll  (IF 

UllANIH  IN  TriE 

[iKIILIN  jAIIHUlCri  AND  THE 

NAuriuAk  Almanac  to  hkuuce  them  to  the  I'uhitionh  fhom  the  I'huvihiunal  Theouy.       1 

Dxte. 

Htilloo«iitria, 

Oenoentrlo.                   1 

n 

Mf'f 

'I'. 

ib 

It 

1830, 

July   24 

—  18.0 

■rinoi 

+   0.0 

—10.3 

+   0.5 

Aug.   13 

18.3 

1007 

0.0 

18.8 

0,5 

Sept.     2 

18.4 

1014 

0.1 

17.0 

0.5 

Bupt.  22 

18.5 

1020 

O.l 

J7.1 

9.4 

Oct.    12 

18.7 

1027 

9.1 

16.7 

0.3 

Nov.     1 

10.0 

1031 

0.2 

16.5 

0.2 

Nov.  21 

—10.3 

+  1038 

+   0.2 

—16.7 

+  0.1 

1831, 

July    19 

_22.5 

+  1112 

+  0.5 

—24.5 

+  10.0 

Aug.     H 

22.0 

II 1 0 

0.5 

23.7 

10.0 

Au(f.  2S 

22.7 

1123 

0.6 

22.7 

10.1 

8f'pt.  n 

22.8 

1128 

9.6 

21.7 

10.0 

Oct.      7 

22.0 

1133 

9.6 

20.0 

0,9 

Oct.    27 

23.2 

1138 

0.7 

20.6 

9.8 

Nov.  16 

23.5 

1142 

0.7 

20.0 

9.6 

Dec.      6 

—23.8 

4-1140 

+  0.7 

—21.0 

9.5 

1832, 

Aug.     2 

—26.0 

+  1108 

+  0.8 

—28.7 

+  10.3 

Aug.  22 

27.2 

1201 

0.0 

27.9 

10.4 

Sept.   11 

27.4 

1202 

9.0 

26.9 

10.3 

Oct.       1 

27.6 

1 205 

0.0 

26.1 

10.2 

Oct.    21 

27.8 

1208 

10.0 

25.4 

10.2 

Nov.    10 

2H.0 

1200 

10.1 

25.1 

10.1 

Nov.   30 

—28.3 

+  1200 

+  10.2 

—25.2 

+  10.0 

1833, 

July    28 

—31.0 

+  1233 

+  10.2 

—34.5 

+  10.7 

Aug.   17 

32.2 

1236 

10.3 

33.7 

10.8 

Sept.     6 

32.4 

1240 

10.3 

32.7 

10.8 

Sept.  26 

32.6 

1242 

10.3 

31.8 

10.7 

Oct.     16 

32.0 

1245 

10.3 

31.0 

10.5 

Nov.     6 

33.2 

1247 

10.4 

30.4 

10.4 

Nov.  25 

—33.5 

+  1250 

+  10.4 

—30.2 

+  10.3 

1834, 

July   23 

— 3S.0 

+  1264 

+  10.5 

—41.2 

+  11.0 

Aug.   12 

38.4 

1266 

10.5 

40.7 

HI 

Sept.     1 

38.4 

1268 

10.6 

30.6 

11.1 

Sept.  21 

38.7 

1270 

10.6 

38.6 

11.1 

Oct.     11 

30. 1 

1275 

10.6 

37.7 

10.0 

Oc.,.    31 

30.4 

1277 

10.6 

36.9 

10.8 

Nov.   20 

30.7 

1282 

10.7 

36.6 

10.7 

Dec.    10 

—40.1 

+  1282 

+  10.7 

—36.1 

+  10.5 

1835, 

July   18 

—  13.0 

+l.^oo 

+  10.8 

—47.7 

+  11.3 

Aug.     7 

44.5 

1311 

10.7 

47.6 

11,3 

Aug.  27 

44.5 

1313 

10.7 

46.6 

11.3 

Sept.   16 

45.1 

1316 

10.7 

45.8 

11,2 

Oct.      6 

45.4 

1318 

10.8 

44.8 

11,2 

Oct.    26 

45.0 

1321 

10.7 

44.0 

10.9 

Nov.    15 

46.5 

1324 

lO.S 

43.6 

10.8 

Dec.      5 

—46.6 

+  1327 

+  10.7 

—  42.5 

+  10.6 

1836, 

July    12 

—51.0 

+  1361 

+  11.0 

—.55.3 

+  11.4 

Aug.     1 

51.3 

l:iC.4 

n.o 

55,2 

11.5 

Aug.   21 

51.5 

1365 

11.1 

54.6 

11.7 

Sept.   10 

—51,0 

+  1368 

+  11.0 

—53.7 

+  11.6 

162 


TUE   ORBIT    OF   URANUS. 


CoBRECTiONH  TO  BE  APPLIED  TO  THE  POSITIONS  OF  UiiANUs — Continued. 

Dr.t«. 

Heliocentric. 

Geouentrio.                    1 

hJ^ 

Mfip 

¥ 

u 

hh 

It 

II 

II 

II 

i83r>, 

Sept.  30 

—52.2 

+  1370 

+  11.0 

—52.4 

+  11.5 

Oli.    20 

52.9 

1372 

11.0 

51.0 

11.3 

Nov.     9 

53. 1 

1375 

no 

.50.5 

11.1 

Nov.   29 

53.9 

1377 

11.0 

.50.4 

11.0 

I>fC.    19 

—03.8 

+  1380 

+11.1 

—49.8 

+  10.9 

1831, 

July      7 

—58.3 

+  1387 

+  11.0 

—02.9 

+  11.4 

July    27 

MA 

i:!87 

11.0 

02.9 

11.5 

AuK-   Hi 

59.2 

13.88 

11.1 

03. 1 

11.7 

Sept.     5 

59.2 

1389 

11.1 

02.0 

11.7 

S.-pt.  25 

00.0 

1391 

11.0 

01.3 

11.5 

Oct.     15 

00.0 

1.393 

11.0 

59.0 

11.4 

Nov.     4 

fil.O 

1394 

11.0 

59.1 

11.2 

Nov.  24 

01.3 

J  394 

11.0 

58.2 

11.0 

Deo.    14 

—02.1 

-}  1393 

+  11.0 

—57.1 

+  10.9 

1838, 

Auff.    11 

— fifi.4 

+  1395 

+11.1 

—71.1 

+  11.0 

Aug.  31 

07.2 

1390 

11. 1 

71.0 

11.7 

Sept.  20 

07.7 

1395 

11.1 

70.2 

11.7 

Oc-t.     !0 

08.1 

1392 

11.1 

08.9 

no 

Oct.     30 

08.5 

1390 

11.1 

07.0 

11.4 

Nov.    19 

08.9 

1388 

11.1 

00.4 

11.2 

Dec.      9 

09.4 

1388 

11.1 

05.8 

11.1 

Dec.    20 

—09.7 

+  1389 

+  11.2 

—05.4 

+  11.0 

1839, 

Aug.     0 

—74.7 

+  1382 

+11.1 

—80.0 

+  11.0 

Aug.  20 

75.1 

1381 

11. 1 

79.7 

11.7 

Sept.  15 

75.5 

1380 

no 

79.0 

11.0 

Oct.      5 

70.0 

1379 

11.1 

77.9 

no 

Oct.     25 

70.4 

1379 

11.1 

70.5 

11.5 

Nov.    14 

70.8 

1379 

111 

75.2 

11.3 

Dec.      4 

77.2 

1378 

11.1 

74.0 

111 

Dec.    24 

77.0 

1377 

11.1 

73.5 

10.9 

1810, 

Jan.    13 

—78.0 

+  1370 

+  11.0 

—73.1 

+  10.7 

June  24 

—8'. 9 

+  1370 

+  11.0 

— Sfi.O 

+  11.1 

Aug.  20 

^•z.i 

1374 

no 

M7.;; 

11.5 

Sept.    9 

83.4 

1370 

11.0 

H7.S 

11.0 

Sept.  29 

83.5 

1377 

n.o 

80.4 

11.5 

Oct.    19 

84.2 

1370 

no 

85.5 

11.4 

Nov.     8 

84.4 

1370 

no 

83.5 

11.3 

Nov.   13 

85.3 

1377 

no 

82.7 

11.1 

Dec.    13 

H5.5 

1378 

n.o 

HI, 5 

10.9 

1841, 

Jan.      7 

—80. 1 

+  lu-7 

+  10.9 

—SI. 3 

+  10.7 

June   10 

-89.8 

+  1378 

+  10.8 

—93.2 

+  10.8 

Aug.   15 

91.1 

13S4 

10.8 

97.0 

11.3 

Sept.     4 

91.5 

13H5 

10.8 

90.9 

11.4 

Sept.  24 

91.9 

1388 

10.7 

90.1 

11.3 

Oct.     14 

92.3 

1 390 

10  7 

94.8 

11.2 

Nov.     3 

92.9 

1392 

10.7 

93  4 

11.1 

Nov.   23 

93.4 

1393 

10.8 

91. H 

10.9 

Dee.    13 

93.8 

I.;95 

10.7 

90.5 

10,7 

18-i'2, 

J  III).      2 

94.3 

1  .•19X 

10  s 

S9.7 

IIJ.O 

I'eli.     11 

—95.0 

-f  U04 

4  10.7 

— S9.0 

+  10.3 

\\ 


« 


THE   ORBIT   OF   URANtJS. 


163 


CouREtmoNs  TO  BE  AppuKD  TO  TiiK  POSITIONS  OF  Uranvs Coiilinued 

Date.                                                     Hellootmtrlo. 

OKOceiitrlc.                   1 

h?. 

V^p 

^i 

11 

ih 

/' 

'  ft 

ft 

II 

1842, 

Tunc   11 

—  Of).  7 

+  1409 

+  l'.'.(i 

—  9S.5 

+  lii.5 

Aiiir.  10 

!llt.l 

14(tS 

10.  r, 

105.3 

10.0 

jVii)f.  ;i() 

y.t.c 

1409 

10.7 

105.8 

10.2 

Sept.    lit 

91t.9 

1409 

,M).(J 

105.2 

10.2 

Oct.     a 

l(tO..T 

1410 

10.5 

104.2 

10.0 

Got.    2!) 

lltU.'J 

1412 

10.5 

102.6 

10.9 

Nov.    18 

.''11.  a 

14ia 

10.5 

100.8 

10.8 

Doc.      8 

l(tl.8 

1414 

10.5 

99.:' 

10.  (i 

D.'c.    28 

1(12.2 

1415 

10.5 

9S.0 

10.4 

iH4a, 

Jul).    17 

-1(12.5 

■tl417 

-f  10.4 

—  97.2 

+  10.2 

All  jr.    ft 

—107.3 

+  1422 

+  10.3 

— M3.7 

+  10.7 

.Auji.  2.'> 

i!!-.- 

1420 

10.2 

114.4 

10.7 

Sept.   14 

Kts.O 

1421) 

'O.l 

114.3 

10. (i 

Oct.      4 

lOX.f) 

1  420 

10.1 

113.4 

10.  fi 

Oct.     24 

lOil.O 

1419 

10.1 

112.1 

10.5 

Nov.   l:} 

Kill. 4 

1418 

10.1 

110.2 

10.4 

Dec.     a 

IIO.O 

1118 

10.1 

lOsf, 

10.3 

Dec.    2;j 

HO.4 

1417 

10. 1 

lOfi.9 

10.1 

1844, 

Jiiii.    12 

—  llit.S 

-i  1418 

+  10.0 

—105.9 

+   9.8 

July    ;iO 

-11.-..1 

-t  1  40C. 

+  '•'( 

—  121.2 

~f  10.0 

Aiijjr.  lit 

II.V4 

IKia 

'.1.8 

122.3 

10.2 

i^i\>{.    s 

II. "..8 

1  too 

!l,S 

122.7 

10  3 

Sept.  2s 

in;,  a 

ia;i7 

9.8 

■    2.4 

10.  a 

Oil.     ix 

licit 

l.'illC. 

:i.7 

i.  ..3 

10.2 

Nin       " 

117.4 

iaii2 

o.c, 

119.7         1             iO.O 

Nov.   27 

1 17.8 

1  .iS9 

o.c, 

117. (!         !               !I.S 

Dec.    17 

lis. a 

ias4 

9.(! 

115.7        I              9.7 

1845, 

i1  II II .        li 

IIS.I! 

lasi 

'.1.5 

111. 3                      9.4 

Juii.    2(< 

—  llll.l 

+  ia78 

+    9.(1 

—  1 13.3                +   9.4 

A  UK.   14 

--12a.-) 

+  1340 

+  9.1 

—  130.4                +   9.4 

J^cpt.     a 

i2a.!» 

laaa 

9.4 

l:!l.4                    9.8 

Sept.  2;t 

124.4 

la.ao 

9.4 

131.7 

9.9 

Oct.    l;i 

124.9 

ia27 

9.3 

130.(i 

9.8 

Nov.     2 

I2r),a 

ia22 

9.3 

129.0 

9.7 

Nov.   22 

12.-).  7 

iai8 

9  2          1 

127.0 

9  5 

Dec.    12 

12(1.2 

laia 

9.2 

125.0 

9.3 

184P., 

.lull.         1 

l2i;.7 

iao9 

9.2 

123.2 

9.2 

Jan.    21 

—  127.! 

+  iaoa 

+  9.2 

—  121.9 

—  9.0 

A  up.  2'.i 

— ia2.o 

-1  125a 

+  8.8 

—1 311.5 

+   9.2 

.Sept.  18 

ia2,a 

1249 

8.8 

l;!9.9 

9.2 

■. 

Oct.       8 

ia2.8 

124(i 

8,8                lao.s 

9.3 

Oct.     2H 

iaa,2 

1242 

8.7         1         ias.4 

9.1 

Nov.    17 

\X).h 

I2a8 

8.7                     13(1.4 

9.0 

Dec.      7 

lai.i 

I2a4 

8.7                     131.4 

8.9 

Dec.    27 

lai.d 

i2aa 

H.C.                     132.3 

S.7 

184T, 

Jiin.     ID 

— lai.'j 

4  i2ao 

+  S.O          1     —130.5 

+    8.5 

A  up.  24 

— lait.r) 

+  1197 

+  8.3              —  ltd  7 

-1    :«.fi 

Sept.   l:! 

11(1.0 

11114 

S.3                   14S.0 

8.7 

Oct.       .! 

'4(i.r> 

mil 

8.3          1         HS.3 

8.7 

Oct.    2;! 

— ilO.9 

+  11^8 

+  8.2                -147.4 

+  8.6 

ao      May,  1 

B73. 

154 


THE   ORBIT  OP  URANUS. 


Ill 


Corrections  to  bb  Applied  to  tub  Positions  op  Uban 

ua — Continued. 

D«te. 

Ileliocentrio. 

Oeooentrlc. 

67i 

tt 

Mh^ 

u 

II 

hb 

It 

184T, 

Not.  12 

—141.4 

-fll85 

+8.2 

—145.8 

+8.5 

Dec.      ■^i 

141.0 

11 83 

8.1 

143.7 

8.3 

Due.    i2 

142.4 

1182 

8.1 

141.4 

8.2 

1843, 

Jan.    11 

_l42.8 

-fll82 

+8.1 

—139.3 

+8.1 

Sept.     7 

—147.0 

+  1105 

+  7.6 

—150.0 

+7.9 

Sept.  27 

14S.:( 

1100 

7.0 

150.7 

8.0 

Oct.     17 

14M.7 

1104 

7.0 

150.4 

8.0 

Nov.     6 

140.a 

1104 

7.0 

155.3 

8.0 

Nov.   2(! 

140.8 

1103 

7.0 

1.53.2 

7.9 

Doe.    U\ 

150.2 

1102 

7.0 

150.7 

7.8 

1849, 

.Full.      5 

\■^{^^^ 

1101 

7.5 

148.3 

7,5 

Jan.    25 

—  151.3 

+  1101 

+7.5 

—140.0 

+7.4 

Sept.     2 

—155.8 

+  1149 

+7.0 

—104.0 

+7.3 

Sept.  22 

150.2 

1150 

7.0 

105.1 

7.3 

Oct.     12 

15fi.5 

1149 

7.0 

105.1 

7.4 

Nov.     1 

150.8 

1148 

7.0 

104.0 

7.4 

Nov.   21 

157.4 

1149 

7.0 

102,4 

7.3 

Deo.     1 1 

157.9 

1148 

0.9 

100.0 

7,1 

Dee.    .tl 

I5S.3 

1U7 

0.8 

157.5 

0.9 

1850, 

Jan.    20 

—  158.0 

+  1147 

+0.8 

—155.0 

+  0.8 

AiiR.   28 

— lo;!.3 

+  1135 

+0.2 

—170.0 

+0.4 

Sept.   17 

103.7 

11. {3 

0,2 

172.0 

0.5 

On.      7 

104.2 

1131 

0.2 

173,3 

0.5 

Oet.     27 

104.7 

1129 

0.1 

173.1 

0.4 

Nov.    1(( 

105.2 

1127 

0,1 

171.8 

0,4 

Dee.      G 

105.6 

1120 

0.1 

109.4 

0.3 

Dec.    2fi 

100.0 

1127 

0.0 

100,7 

0.2 

1851, 

Jan.    15 

100.3 

1127 

0.1 

103,8 

0.1 

l\\>.      4 

—100.7 

+  1124 

+0.0 

—101.8 

+  5.9 

Sept.   12 

—  171.5 

+  1109 

+5.5 

—180.0 

+5.7 

Oet.       2 

171.9 

11  Of 

5.5 

181.4 

5.8 

Oet.     22 

172.4 

1103 

5.5 

181,0 

6.8 

Nov.    11 

172.9 

1103 

6.4 

180.8 

5.7 

Dee.       I 

173.3 

1103 

6.3 

178,7 

6  5 

Dee.    21 

173.8 

1102 

5.2 

170.1 

5.3 

185-2, 

Jan.    10 

174.3 

1101 

6.2 

173.4 

6.3 

Jan.    30 

—174.7 

+  10U8 

+5.1 

—  170.2 

+5.1 

Sept.     0 

—  179.0 

+  1074 

+  4.7 

—180.8 

+4.9 

Sept.  2(; 

179.4 

1073 

4.0 

188.9 

4.8 

Oet.     l(i 

179.8 

1071 

4.5 

180,7 

4.7 

Nov.     5 

1H(».3 

1008 

4.5 

180,0 

4.7 

>■•  ■'.   25 

180.7 

1007 

4.4 

187,8 

4.6 

Ito.    15 

181.2 

1005 

4.4 

185,3 

4.6 

185.1, 

Jan.      4 

181.5 

1004 

4.4 

182,3 

4.5 

Jiin.     24 

181,8 

1002 

4.4 

179.0 

4.4 

Fob.    l;i 

—  182.1 

+  1000 

+4.3 

—170.9 

+  4.2 

Sept.      1 

—185.8 

+  1055 

+3.9 

—192.4 

■14.0 

Sept.   21 

180,2 

1055 

3.8 

195.2 

3,9 

Oet.      1 1 

—  ISC. 5 

+  1055 

+3,7 

—190,7 

+  3,9 

THE 

ORBIT  OF 

URANUS. 

165 

CoRiiECTioNS  TO  BK  Api'UED  TO  THE  POSITIONS  OP  Ukanus — Continued. 

Dat«. 

Heliooentrio. 

Oaooentrio.                 1 

ft 

J/Sp 

'i 

hi 
It 

1853,    Oct.    31 

—isfi.s 

+  1054 

+  3.fi 

— 19fi.8 

+3.8 

Nov.    20 

1M7.1 

1054 

3.5 

195.fi 

3.7 

Dec.    10 

1H7.5 

10f>» 

3.5 

193.6 

3.B 

Dec.    30 

1«7.9 

1053 

3.5 

190.7 

3.6 

1854,    Jnn.     19 

l«S.l 

1053 

3.5 

187.4 

3.5 

Feb.      8 

—188.5 

-t-1052 

+  3.5 

—184.5 

+  3.5 

Sopt.  Ifi 

—192.2 

+  1034 

+2.9 

—200.5 

+  3.U 

Oct.      f, 

i92.(; 

1032 

2.8 

202.9 

2.9 

Oct.    2(5 

193.0 

1032 

2.7 

203.7 

2.8 

Nov'.    IS 

193.3 

1032 

2.r, 

203.0 

2.7 

Dec.      5 

193.7 

1020 

2.fi 

201.3 

2.7 

Doc.    25 

194.0 

1025 

2.5 

198.fi 

2.« 

1855,    Jan.     14 

194.2 

1021 

2.5 

195.2 

2.5 

Fclj,      3 

—194.5 

+  1018 

+2.4 

—192.0 

+  2.4 

Oct.       1 

—  198.3 

+  1010 

+  1.8 

—207.9 

41.9 

Oct.    21 

19S.7 

1009 

1.7 

209.fi 

1.8 

Nov.    10 

199.0 

1008 

in 

209.7 

1.7 

Nov.  30 

199.3 

1007 

1.5 

208.4 

l.fi 

Doc.    20 

199.fi 

loofi 

1.5 

20fi.O 

l.fi 

185i!,    Jnn.      !) 

199.8 

1005 

1.5 

202.fi 

1.5 

Jan.    2!) 

2(10.1 

1003 

1.4 

199.2 

14 

Feb.    18 

—  200.3 

+  1001 

+  1.3 

— 19«.l 

+  1,3 

Oct.     1.^ 

—203.3 

+   9(12 

+0.7 

-213.9 

+  0.7 

Nov.     4 

203.7 

9.-.  7 

0.7 

214.9 

0.7 

Nov.   24 

203.9 

953 

0.7 

214.5 

0,7 

Dec.    14 

204.3 

949 

O.fi 

212.4 

0,fi 

185T,    Jan.      3 

204.4 

945 

O.fi 

209.3 

O.fi 

Jnn.    23 

2(14.5 

941 

0.5 

205.  (; 

0.5 

Feb.    12 

—204.7 

+   'J34 

+0.4 

—202.2 

4  0.4 

Sept.  20 

— 20fi.9 

+   879 

—0,2 

—214  0 

—  0.2 

Oct.     10 

207.2 

87  4 

0.3 

217.1 

0.3 

Oc't.     30 

207.5 

808 

0.3 

2  is,  8 

0.3 

Nov.  in 

207.8 

Kfil 

0.4 

219,0 

0  4 

Dec.      fl 

208.0 

853 

0.4 

217,fi 

0.4 

Dec.    20 

20S.1 

840 

0.5 

215,0 

0.5 

1858,     Jim.    IS 

20S.2 

S39 

O.fi 

211,5 

O.fi 

Fell.      7 

20S.3 

830 

O.fi 

207,7 

0  fi 

Feb.    27 

—208.4 

+   823 

—O.fi 

—204,2 

—O.fi 

Oct.      f) 

—210.4 

4    741 

—  13 

—219,1 

—1.3 

Oct.     25 

210.fi 

732 

1.4 

221,4 

1.5 

Nov.    14 

210.8 

723 

14 

222.2 

1.5 

Dec.      4 

211.0 

713 

1.5 

221.9 

l.fi 

Dec.    24 

211.2 

704 

1.5 

219,7 

l.fi 

1859,    Jan.    13 

211.3 

095 

1.5 

21fi,fi 

1.5 

Feb.      2 

211.4 

«8fi 

l.fi 

213,0 

l.fi 

Feb.    22 

— 2I1.« 

+  fi77 

—  15 

— 209,4 

—1.5 

Oct.     20 

—213.1 

4    57 fi 

—2.1 

222,  s 

—2.2 

Nov.      9 

213.2 

509 

2.1 

224.5 

2  2 

Nov.   29 

—213.4 

1          +   5fil 

—2. 1 

—224.8 

—2.2 

156 


THE   ORBIT  OF   URANUS. 


CuaBECTKlNH  TO  UK  A 

"i'LiED  TO  TUB  TuMiTioNH  OK  Uranvd — Conlinuetl. 

Date. 

Uelioconlrie. 

OttovcDlrle.                  1 

hX 

Sifif 

¥ 

hi 

,M 

11 

It 

it 

If 

185», 

nrc.    19 

— 2I3..'> 

-f554 

—2.3 

—223.5 

—2.3 

1»UU, 

.lull.       8 

2i3.r> 

54fi 

2.3 

220.  d 

2.3 

.Inn.    28 

213.(5 

539 

2,3 

217.4 

2.3 

Fob.    17 

—213.7 

+533 

—2.3 

—213.5 

—2.3 

Sept.  24 

—214.7 

-f  45fi 

—2.8 

—210.7 

—2.9 

Oft.     14 

214.8 

451 

2.8 

223. 1 

2.9 

Nov.     3 

:      0 

444 

2.n 

2-.?5.0 

3.0 

Nov.   2H 

21;'..  1 

437 

2.9 

220.8 

3.1 

Dec.     1H 

215.3 

431 

3.!) 

220.3 

3.3 

1801, 

.lull.      2 

2ir..3 

424 

3.1 

223.8 

3.3 

Jim.    22 

2ir,.3 

417 

3.1 

221.0 

3.3 

Fib.    12 

-215.3 

+  411 

—3.1 

—217.3 

—3.1 

Oct.    29 

—215.fi 

+  ;!3!) 

—3.7 

—225.0             -3.9           1 

Nov.    13 

2l5.(i 

333 

3.8 

220.9 

4.0 

Dec.      8 

215.7 

32!) 

3.8 

227.4 

4.0 

Dw.    2s 

215.8 

321 

.1.8 

220.3 

4,0 

1 80  2. 

.Ian.    17 

215,8 

320 

3.9 

223,5 

4.0 

l'\-l).      (i 

215.8 

315 

4.0 

220,0 

4  1 

Ffl).    2(! 

—215.!) 

4-310 

—4.0 

—210.0 

—4.0 

Oct.    24 

— 21fl.O 

424(5 

-44 

—224.2 

— 4.fi 

Nov.    i:{ 

21t;.0 

241 

4.5 

220.0 

4  7 

Dc,       3 

21f..O 

237 

4  5 

227.7 

4.7 

Dec.     23 

215.0 

2;!2 

4.0 

227.3 

4.8 

lues, 

.litri.     12 

215.8 

22ti 

4.7 

225.0 

4.9 

F«-l>.      1 

J 15. 8 

21!) 

4  7 

221.8 

4.8 

Fob,    21 

415.7 

213 

4.7 

218,1 

4.8 

Mur.    13 

-215.7 

4-208 

—4.8 

—214,1 

—4.8 

Nov.     8 

—215.3 

4  13!» 

—5,3 

—224.fi 

—5  5 

Nt)V.    28 

215.2 

133 

5,4 

220.5 

5.7 

l>.M'.       18 

215.1 

n;fi 

.'.,4 

220.8 

5.7 

18G4, 

.litii.      7 

215.(1 

120 

."1.5 

225,4 

5.8 

.lull.    27 

214  !t 

114 

.'■.,5 

222,7 

5,7 

Feb.    ii; 

214.8 

108 

5,fi 

210,3 

5,7 

Mtticb  7 

— 2U.8 

4-103 

—5,0 

—215.3 

— 5,fi 

Oct.     13 

—213.8 

4    21 

-fi,0 

—217.8 

—0,1 

Nov.     2 

213.7 

Ifi 

0,0 

221.3 

0,3 

Nov.  22 

21.3.7 

!) 

0,1 

2  2  to 

0,4 

Dec.    12 

213.fi 

4-     1 

0.2 

225,1 

0.5 

1865, 

.lull.        1 

213.4 

—    (1 

0,2 

224  7 

0.5 

•Ian.    21 

213.2 

14 

0,3 

222,7 

0,5 

Feb.    10 

213  1 

20 

0,2 

2?0.fi 

0,4 

March  2 

—213.0 

—  27 

—0,3 

—215.7 

—0.4 

Oct.      8 

—2ll.fi 

—103 

—0.8 

-213,5 

—0,9 

Oct.    28 

211.4 

no 

fi.S 

217,2 

7,0 

Nov.    17 

211.3 

117 

0,9 

220,0 

7,2 

Pec.       7 

211.1 

124 

0,9 

222,0 

7.3 

Dee.    27 

210.!) 

132 

r,!» 

222  4 

7.3 

laort, 

.lull.    l*t 

21(1  7 

130 

7,0 

r,-M  5 

7.4 

Feb.      5 

•'10.fi 

i;5 

(I.O 

218,8 

7.3 

Feb.    2.''> 

210.4 

151 

7.0 

215.4 

7  3 

Mur.    17 

—210.1 

—158 

—7  1 

-211.2 

—7  1 

f  ^ 


THE   OUUIT   OF   URANUS. 


167 


CiiKRECTidNS  TO  iiE  Ai'Pi.iEU  TO  THE  I'oHiTiONS  OF  I'uAMs — Continued. 

Date. 

iI«lior«ntrio. 

Oeocentrlo.                   1 

hx 

II 

J/^p 

ft 

hh 

II 

l8Cf., 

Oft. 

3 

—208.2 

225 

—  7.4 

—207.3 

—  7.4 

Oct. 

2.1 

20S.1 

2.30 

7.4 

211.5 

7.0 

Nov. 

12 

207.9 

2.35 

7.5 

214.9 

7.8 

nee. 

2 

207.8 

241 

7.fi 

21 7. « 

8.0 

Dee. 

22 

207.fi 

24« 

7. ft 

218.8 

8.0 

ISf.T, 

.lull. 

11 

207.4 

251 

7.fi 

21S.5 

8.0 

Jan. 

:tl 

207.2 

25fi 

7.7 

21(1.7 

8.0 

F.I,. 

20 

207.0 

2(i2 

7.7 

213.8 

7.9 

Mar. 

12 

—20(1.7 

—  208 

—  7.7 

—200.8 

—  7.8 

Nov. 

27 

—203.2 

—  351 

—  8.2 

—211.4 

—  8.0 

Dec. 

17 

203.0 

358 

8.2 

213.5 

8.0 

1808, 

•la  II. 

(> 

202.8 

3(15 

8.3 

213.7 

8.7 

•Ian. 

2(i 

2(t2.4 

371 

8.3 

212.7 

8.7 

Fell. 

la 

202.2 

37(! 

8.3 

210.3 

8  0 

Miir. 

t> 

201.8 

:',s-2 

8.4 

207.0 

8.0 

iMar. 

2I-. 

—201.4 

—  387 

—   8.4 

—202  0 

—  8.4 

O.t. 

12 

—  lOH.O 

—  45S 

—   S.C, 

—  107.2 

—  8.0 

XnV. 

1 

107.8 

4(15 

8.7 

201.1 

8.9 

N..V, 

21 

107.5 

473 

8.8 

203. 8 

0.1 

Kit. 

11 

107.2 

4HI 

8.8 

20(1.2 

0.2 

I».r. 

::i 

ioi;.8 

400 

8.8 

207.5 

0.3 

18(1  it, 

.Ian. 

20 

1!h;.4 

4!I8 

8.8 

207.1 

0.3 

Fell. 

ii 

loco 

507 

8.8 

205.4 

0.2 

Mar. 

1 

lO.'i.C. 

515 

8.0 

202.4 

0.2 

Mar. 

21 

-10,').  3 

—  524 

—  8.;) 

—  l!is.4 

—  0.0 

Deo. 

(i 

—1 110.0 

—  107.0 

Dee. 

2<i 

IHO.5 

los.o 

1S70, 

.Ian. 

ir> 

IH'.I.O 

—  (1(15 

—  0.4 

100.5 

—  9.9 

Fell. 

14 

ISH.IJ 

<17(> 

0.4 

10H.7 

0.0 

Feb. 

24 

ISM  0 

fiSCi 

0.4 

10(1.2 

0.8 

Mar. 

ir, 

1S7.5 

ri07 

0.4 

103.0 

9.0 

April 

;'» 

—  1H7.2 

—  708 

—  0.4 

-1x0. 4 

—  0.4 

De.v 

1 

— isi.r. 

—  S4(! 

—  O.fi 

—  1.S(!.5 

—  10.0 

Dee, 

21 

isl  1 

m5'.» 

O.fi 

IHO.O 

10.1 

1871, 

•Ian. 

It 

l.SO.fl 

K72 

0.7 

100,3 

10.2 

.Ian. 

30 

170.S 

8.M3 

0.8 

100.1 

10.3 

Fc'l.. 

10 

no  1 

.X04 

0.7 

1X8.3 

10.1 

Mar 

11 

17X  I 

OOli 

0.7 

1M5.(! 

10.0 

Mar. 

31 

—  177.0 

—  '.110 

—  0.8 

—  181.2 

—  0.9 

Dee. 

H! 

—  171.C. 

—  los.l 

—  10  0 

—177.4 

—  10.4 

IH'i-i, 

.Ian. 

r. 

171.0 

1005 

10. 1 

170.3 

10.0 

Jan. 

25 

170. 5 

1107 

lO.U 

HO.  2 

10.0 

Fell. 

14 

100.8 

1 1  20 

10.0 

170.(1 

10.5 

Mar. 

r. 

100.3 

1133 

10.1 

177.8 

10.5 

Mar. 

2.*» 

l(l.s.7 

1115 

10. 1 

174.0 

10.3 

Apri 

It 

IC.M.I 

11511 

10.1 

171.3 

10  1 

May 

4 

—  1(17.(1 

—  11(18 

— lo.l 

—  1(1(1.3 

—  10.0 

■fli 


V0 


158 


THE   0  lUU  T   O  r   URANUS. 


CIIAPTEll   VII. 


HI 


FORMATION  AND  SOLUTION  OF  THE  KQUATIOXS  OF  CONDITION  RESULTING 
FROM  THK  riUICKDlNU  COMTARISONS. 

In  the  preceding  cliaptcr  wo  have  ohtaiiicd  from  observations  a  series  of  cor- 
rections to  the  {geocentric  positions  of  Vrunns  resulting  from  the  provisional 
theory.     The  furUier  operations  are  as  follows:  — 

1.  To  reduce  all  the  corrections  in  right  ascension  and  declination  to  correc- 
tions in  geocentric  longitude  and  latitude.  Most  of  the  corrections  arc  already 
so  expressed,  so  that  this  reduction  is  necessary  in  only  a  few  cases. 

2.  To  find  the  mean  value  of  the  correction  in  geocentric  longitude  during  each 
opposition,  and  to  express  this  mean  value  in  terms  of  tiie  correction  to  the  helio- 
centric cf)-ordinates. 

;{.  To  express  these  corrections  to  the  heliocentric  co-ordinates  in  terms  of  cor- 
rections to  the  elements  of  Uranus  and  the  mass  of  Neptune. 

4.  To  solve?  the  equations  of  condition  thus  formed. 

The  first  of  these  jjroeesses  is  too  sinii>le  to  mak(>  it  necessary  to  present  any 
details  of  it.  With  regard  to  the  second  1  have  sought,  not  the  simple  correction 
to  the  geoeentri*:  longitude,  hut  this  correcticm  multiplied  'jy  smh  a  factor  as  it 
was  supposed  would  make  the  i)rol)ahle  error  of  tin-  correction  0".;").  The  equations 
for  expressing  the  error  of  geoc(>ntric  longitude  in  terms  of  trrors  of  heliocentric 
longitude  and  radius  vector  have  been  given  on  page  I'i!).  The  first  observation 
of  I'lamstead,  p.  107,  gives  the  ('(piation 

4- 2v>'=1.04.U  +  .n'27,^p 

h?.  being  the  correction  to  the  h<lif)centric  longitude,  and  f,p  that  to  the  Neperian 
logaritlim  of  tin;  radius  vector.  From  the  discordance  of  I'lamstead's  clock  errors 
it  nniy  l»e  estimated  that  the  probable  error  of  the  first  member  of  this  ecpuition  is 
10".     Therefore  we  divide  the  eepiation  by  20,  which  gives 

jV?  =  I'M  =  .0ry2,U  +  AWUp. 

In  the  op])osition  of  171")  we  have  four  observations.  The  best  weve  those  of 
March  4  and  10,  of  which  wt'  may  estimate  the  ])robable  error  at  10",  anu  the 
worst  that  of  March  5,  of  which  the  i)r()babl(>  error  may  be  estimated  at  20", 
while  that  of  April  2!)  is  intermediate  in  certainty.  1'he  separate  obscTvations 
give  the  e<piations 

\MftX;  AVeight,  4 

1.06^3,;  "Weight,  1 

l.OfxSX;  "Weight,  4 

1.04V;i-f  .04V:  Weiglit,  2. 


March    4,  ,V  =  -f-2.S" 
March    T),  ,V  =  -f-  •*  ^ 
March  10,  ,V^  + W 
April    2!),  <V=-f-    - 


Mean 


hl—-{-  27.G  =  l.OiJtJAX  -f.OOa  V ;  probal)le  error  =  ±  6*. 


.^ 


THE   ORBIT   OF   URANUS. 


169 


Applying  tlic  correction  — V.l  for  equinox,  anil  dividing  by  12,  the  equation  of 
condition  becomes 

^\^fl  =  +  2".2  =  ().0885;i. 

In  tliis  way  the  following  equations  were  obtained.  It  is  deemed  unnecessary 
to  give  tlic  details  of  the  process,  as  it  is  one  which  every  one  can  go  over  for  himself 
from  the  data  already  given,  and  can  reproduce  all  the  resjdts,  except  so  far  as 
they  depend  on  the  relative  weights  assigned  to  the  difl'cnint  groups  of  observations 
during  one  and  the  same  opposition. 


No. 


Date. 


Equations. 


Number  of 

obHvrvatiuus 

iu  U.  A. 


1 

1G91.0; 

^o^i 

=  + 

1.1  =  .052r^J 

.+ 

.001<^p 

1 

2 

171 '1.2 

vi 

=-■  + 

2.2  =  .088 

4 

3 

1748.8 

k 

=  + 

12.8  =  .338 

+ 

.017 

1 

4 

1750.8 

h 

=  + 

11.8  =  .345 

+ 

.010 

3 

6 

1153.9 

h 

=  + 

ll.G  =  .333 

+ 

.016 

1 

6 

n5G.7 

I 

=  + 

5.0  =  .210 

+ 

.003 

1 

7 

17G9.0 

! 

=  + 

4.8  =  .203 

+ 

.010 

8 

8 

1782.0 

,} 

=  4- 

3.0  =1.370 

21 

9 

1783.0 

1 

=  + 

1.25  =  1.030 

— 

.002 

13 

10 

1784.0 

1 

=  + 

1.92=1.026 

— 

.008 

13 

11 

1785.0 

1 

=  — 

0.2G  =  1.034 

+ 

.006 

10 

12 

1788.0 

i 
.7 

=  + 

1.23  =  0.084 

+ 

.006 

5 

i:j 

1789.0 

J 

=  + 

1.58  =  0.504 

+ 

.008 

6 

14 

1  "190.0 

i 

=  — 

0.52  =  0.514 

— 

,013 

4 

15 

1791.0 

.1 

z=  

O.'oG  =  0.G84 

— 

.010 

7 

IG 

1792.0 

h 

=  — 

0.12  =  0.340 

— 

.011 

3 

17 

1193.0 

J 

=  _ 

0.19=0.512 

— 

.015 

5 

18 

1794.0 

h 

=  + 

0.79  =  0.344 

— 

.006 

3 

19 

1795.0 

-» 

r=  — 

O.GG  =  0.G83 

— 

.019 

7 

20 

n9G.O 

h 

=  — 

O.G8  =  0.514 

— 

.015 

4 

21 

1797.1 

i 

=  — 

0.35  =  0.528 

3 

22 

1800.2 

h 

= 

0.00  =  0.352 

3 

2;j 

1801.2 

h 

=  — 

().2(>  =  0.352 

2 

24 

1S02.3 

1 

=^  + 

O.S5=r^l.05 

13 

2) 

l.S(t5.:5 

1 

=  4- 

0.(i9  =  1.05 

13 

2G 

1N()(;.3 

i 

=  + 

O.20  =  0..>2 

5 

27 

1801.3 

1 

=  -\- 

2.34  =  1.045 

16 

28 

1808.3 

i 

t=  — 

0.0 1  =  0.52 

6 

29 

1809.3 

1 

-=  + 

1.80=0.70 

9 

30 

1810.3 

1 

-  + 

2.39  =  1.05 

16 

31 

1811.3 

I 

-  + 

1.49  =  1.01 

.^_ 

).0l 

11 

32 

1812.4 

1 

> 

-  + 

O.S  =0.53 

8 

33 

IS  13.  J 

I 

=  + 

1.2  =().5:{ 

9 

34 

1814.4 

1 

-=-h 

1.1  =  1.05 

15 

160 


1 

r  II  K 

OIUMT   OF    UllANUS. 

No. 

Pnle. 

E<|unlioii» 

. 

Xninlier  of 

oltsiTvationa 

ill  It.  A. 

35 

1815.4 

3 

V=+    2.1 

-=1.58,U 

20 

36 

1818.4* 

•J 

=  +    0.3 

=  1.58 

24 

37 

1819.4 

i 

=  —    0.8 

=  1.05 

11 

38 

18-20.5 

1 

=  —    1.3 

=  1.05 

14 

3!) 

18-21.5 

i 
it 

=  +    0.9 

=  0.70 

10 

40 

18:il>.5 

i 

=  +    0.9 

=  0.70 

7 

41 

18-23.5 

.1 

=         0.0 

=  0.70 

11 

4'i 

18li4.5 

1 

=  4-    0.5 

=  1.05 

12 

43 

18-25.5 

■» 

=  —    0.4 

=  0.70 

7 

44 

18-2(5.5 

1 

=  +    0.3 

=  1.05 

11 

45 

18-27.7 

2 

=  -    '2.7 

=  *2.07 

+  0.04  ,^p 

37 

4G 

18-28.7 

-'i 

=  —    *2.7 

=  *2.57 

+  0.07 

67 

47 

18*29.7 

n 

=  -    2.4 

=  '2.59 

-1-  0.04 

61 

48 

1830.7 

n 

=  —    4.9 

=  2.56 

4-  0.07 

73 

49 

1831.7 

2 

=         0.0 

=  *2.oa 

-j-  0.05 

54 

60 

183*2.7 

2 

^^ 2.2 

=  *2.07 

-\-  0.05 

65 

51 

1833.8 

n 

=  —    4.1 

=  *2.58 

4-  0.08 

88 

62 

1834.8 

n 

=  —    3.9 

=  *2.57 

-j-  0.08 

91 

63 

1835.8 

n 

=  —    5.1 

=  *2.59 

-f  0.05 

82 

64 

183(5.8 

3 

=  —    7.0 

=  3.11 

+  0.08 

157 

65 

1837.8 

3 

=  —  3.r. 

=  3.11 

+  0.01 

162 

6(i 

1838.8 

3 

=  —     1.6 

=  3.11 

H-  0.06 

193 

67 

1839.8 

3 

=  -    1.2 

=  3.11 

4-  0.06 

170 

68 

1840.8 

3 

=  —    1.5 

=  3.11 

-f-  0.06 

1*24 

69 

1841.8 

3 

=  +    0.8 

=  3.10 

-f-  0.06 

108 

60 

184*2.8 

3 

=  +    1.6 

=  3.10 

-f  0.06 

169 

61 

1843.8 

3 

=  +    4.7 

=  3.12 

+  0.04 

111 

62 

1844.9 

3 

=  +    5.1 

=  3.11 

+  0.03 

106 

63 

1845.9 

2 

=  +    4.2 

=  *2.08 

+  0.02 

55 

64 

184G.9 

3 

=  +    (5.3 

=  3.10 

4-  0.04 

98 

65 

1847.9 

*2 

=  +    6.8 

=  *2.07 

4-  0.03 

74 

66 

1848.9 

2 

=  +    4.4 

=  *2.()8 

4-  0.02 

59 

67 

1849.9 

2 

=  +    (5.6 

=  *2.08 

4-  <».04 

33 

68 

1850.9 

2 

=  +    7.2 

=  *2.08 

4-  0.01 

46 

69 

1851.9 

2 

=  +    «.3 

=  2.07 

4-  0.04 

42 

70 

185*2.9 

2 

=  +    6.6 

=  '2.07 

+  0.03 

54 

71 

1853.9 

•2 

=  -f    7.9 

=  *2.09 

4-0.01 

49 

72 

1854.9 

2 

=  +    8.9 

::=  2.09 

4-  0.02 

49 

74 

1855.9 

2 

=  -l-    8.5 

=  *2.08 

+  0.04 

48 

76 

1 85(5.9 

2 

=  -f    7.9    . 

=  '2.08 

+  0,04 

45 

76 

1858.0 

n 

=   -f    10.3      : 

^  *2.(;i 

-|-  0.09 

66 

*  The  results  for  ISir,  uikI  IhU  wore  ipiiiillcil  in  this  list  tlirmigli  oversight. 


J 


THE   ORBIT  OF  URANUS. 


161 


Numlirr  of 

No. 

Date. 

EqilMinriR. 

obHiTviitionii 

ir 

ill  It   A. 

77 

1859.0 

2i'' 

/=  + 

10.6 

=        )  ,U  +  0.03  -V 

58 

78 

18G0.0 

•4 

=  + 

8. -2 

—  V'.CiO 

+  0.05 

G4 

79 

IHGI.O 

•_> 

=  + 

(i.:j 

=  2.0!) 

+  o.o;j 

41 

«() 

lS()'i.() 

n 

=  + 

7.0 

=  2.(iO 

4-  0.05 

60 

81 

1H(5:J.0 

n 

=  + 

(>.<» 

=  2.5!) 

-f-  0.08 

88 

82 

ls(i4.() 

') 

=  + 

4.:J 

=  2.0!) 

+  0.0:{ 

35 

8;j 

i.s,;5.o 

H 

=  + 

y.i) 

=  1.57 

4-  0.04 

37 

84 

18()().() 

n 

=  + 

1.1 

=  2.(>0 

4-  0.06 

76 

85 

1S(H.0 

n 

=;  

1.8 

=  2.G0 

+  0.02 

83 

8(5 

isr.s.o 

2 

=  

:j.8 

=  2.0!) 

4-  0.04 

40 

87 

i,s(;!).() 

n 

=:  — 

!).l 

=  2.01 

-f  0.02 

r.s 

8H 

imo.o 

•i 

:^=  

y.o 

=  2.084 

+  0.054 

31 

8!) 

1H71.0 

u 

=:  — 

10.(> 

=  1.500 

-f  0.040 

21 

!)() 

187  2. 1 

n 

==  — 

ii).i 

=  2.000 

-f  0.070 

50 

inihor 

of  ohsiTX 

atioi 

IS  ill  11 

A., 

.     .     .     . 

.     . 

37()3. 

1  -f  0.i»!):J!>  CDS  <j  -\-  0.0055  cos  2y 
'^  =  1.!)!)!)  sin  7  4  0.1 17  sin  2'/  +  0.007  sin  3'/ 
'  =  _  2.000  COS  7  _  0. 1 1 7  co.s  2j  —  0.007  cos  3y 


AVc  have  next  to  express  the  vnlnes  of  A?,  and  .'j»  in  tcnn.s  of  tiic  corn'ctions  to 
the  elements.     Dirt'erentiatinfi;  the  expressions 

■A^l  +  (2/-  —  It')  sin  (/  —  7t)  +  Jfc"  sill  (2/  —  2.n)  +  ]  ^e"  sin  »•»  sin  (3/  —  3:t)  +  etc. 
,,  =  »  -f  !«'  4  (  —  '    }-  y)  'o^  (/  —  7t)  —  y- cos  (2/  —  ''n)  —  etc., 
with  respect  to  /,  ',  and  rr,  and  reducing  tlie  coc  flicients  to  nuniliirs,  we  find 

il 

6v 
d% 
f)n 
We  liavc  lierc  put  /  for  the  mean  longitude,  or 

/  =r  «/  -|-  £ 

whence 

I'll  vl 

Also,  from  the  expression  for  p 
di 

Si' 

cJp  _    cip  _  2^ 

rn         (i       on 
The  values  of  tliese  coefficients  wliich  depend  only  on  g  arc.  shown  in  the  following 
table : 

::i       May,  1873. 


-.;=(' 

i'--(I- 


t 

Jl 

—  l^)  sin  U  +  ■?.'-"  sin  "i'J  + 
\t^)  cos  g  —  |c  cos  2//  —  etc. 


1fi 


1U2 


THE   OUDIT   OF   UUAXU8. 


d-K 

dx 

d?. 

^l 

s^> 

c?p 

9 

6e 
+  1.099 

Oe 

«/.T 

de 

de 

eiln 

0 

—2.124 

0 

—1.00 

u 

1.0!)!) 

+0.0,'!!» 

2,124 

+  0.001 

1.00 

—0.02 

2 

I.01t!» 

0,070 

2. 1 2:t 

0.002 

1,00 

0.0:1 

■A 

l.Ollll 

Oils 

2.121 

0,002 

1.00 

O.O.'i 

4 

l.t»!l!) 

0,1. 'iS 

2.  IIS 

o,oo:{ 

1.00 

0.07 

T) 

+  1.011') 

+0.  |;m; 

—2.114 

+  0,004 

—  1.00 

— 0.00 

0 

I.OD!) 

o,2:i:> 

2.110 

0,00:) 

0.0!) 

u,  10 

7 

1.00!) 

0,271 

2.ior) 

oooc 

0.0!) 

0,12 

« 

l.O'.tS 

0,.ll.l 

2.00!) 

0,001; 

0.0!) 

0,14 

'.» 

roos 

0,:!.'i2 

2.002 

0,007 

0.!l!) 

oin 

10 

^  l.OilS 

-|-o,;ioi 

— 2.0xr> 

+  o,oos 

— O.OS 

—0,17 

11 

1,007 

0  4.iO 

2.077 

0,00!) 

0.  !IS 

0.10 

12 

1.00' 

0  ics 

2.oi;o 

0,010 

0.!I8 

0,21 

l.'l 

1.007 

o,riO.". 

2.o:i0 

0  011 

0.!I7 

0.22 

14 

1.0!ll'> 

0  .'.44 

2,01!) 

UOll 

0.07 

0.24 

i:. 

+  1.0!)(! 

40,, -.si 

— 2,0,iS 

40.012 

—0.07 

— 0.2»1 

li; 

I.OO.'i 

0,1;  IS 

2,027 

0,01:1 

0.!)C 

0.2s 

n 

1.0!»r> 

o,cr,5 

2,014 

OOll 

O.OC 

0.20 

IH 

1.004 

o,r,o:t 

2,001 

o,oi:) 

O.!).") 

o.;ti 

1!) 

1.004 

0,720 

1,0  S7 

o.oir> 

0.!)r> 

0.:!:i 

iiU 

+  1.00.J 

+o.7cr> 

—  1.07:1 

40,01c 

—0.04 

_o.:i4 

21 

1.002 

0,S01 

1.057 

0.017 

0.0:1 

o.;ic 

22 

l.ooi 

o,n;ic 

1,041 

O.OIS 

0.!):< 

o.:i7 

2:1 

1.000 

0,><72 

i,!i2r. 

o,ois 

0.02 

o,:!0 

2. 

1  000 

0,!I07 

1  007 

0,010 

0,01 

0,41 

2". 

4  1  oso 

40,012 

—  is'io 

4  0,020 

— 0,!ll 

—0,4  2 

2ii 

LOSS 

o.07r, 

l,s72 

0,0:>1 

0,00 

0,4  1 

27 

1.0H7 

1, 010 

i,.srp2 

0,021 

o.si) 

0,4.-. 

2S 

l.oHi; 

i,04;i 

l.s:!2 

0,022 

0,HS 

0.47 

2'.> 

i(i><r» 

1,07c 

I,M1 

o,o-j;i 

0.S7 

0  4S 

;ii) 

+  I.0M4 

+  l,los 

—  1,700 

40,02;! 

— 0.S7 

— O.j'iO 

:!l 

I,0S4 

1.140 

1,7C0 

0.0:>4 

o.sc 

o.;'>l 

32 

l.o.s;i 

1  172 

1  717 

o,o2r> 

o.sri 

o.r.:i 

:;;t 

1.0S2 

1,203 

1  721 

0,02c 

0.H4 

o.:.4 

HI 

l.oso 

l,2;!;j 

1,700 

0,02c 

o..s:i 

o.r.c 

;!■. 

-t  1,070 

-1  1,2c  4 

—  l,(i7C 

4  0,027 

— 0.H2 

— 0..-.7 

lie. 

l.o-H 

1,204 

l,r,.M 

0.02H 

0.x| 

O.fiO 

;i7 

1.077 

l,:i2:i 

1.C2H 

0,02s 

O.XO 

o.co 

;iH 

1.07C. 

1  :i:.l 

l,COl 

0,020 

0.70 

o.<:2 

:!!i 

1.071 

I,:i70 

i,r.74 

o.o:;o 

0,7s 

o,<i:i 

Id 

H  l,07:i 

-t  1,407 

—  i,r.is 

4o.(i:;o 

—0,77 

— 0,C4 

\\ 

1,072 

1  4::  1 

i,r,2i 

0,0:;  1 

0,7C 

o.cc 

42 

1,070 

1.4  CO 

1,404 

0  o::i 

0,71 

0.C7 

4:1 

IOC!) 

1  4s(! 

1,4CC 

0  (i:!2 

o.7:t 

O.CS 

(4 

IOCS 

i.r.ii 

1 , 4;!s 

0,0:1:1 

0,72 

0.70 

4:. 

4  l,or,7 

+  l.r.:ic 

—1.400 

40,0:;:! 

—0.71 

—0.71 

4(; 

1,0C5 

i.r.ci 

i,:!so 

o.o:s4 

0.70 

0.71 

47 

1,0c  4 

l.r)S4 

l.:!.M 

o,o;ti 

O.CS 

0.72 

4x 

l,OC.'< 

l.COli 

l.:!2o 

o,!):):* 

o,c7 

0.74 

4!l 

l,ocl 

l.r,20 

1 .  200 

o,o::c 

o.cc 

0.7'> 

.Ml 

-f  l,oco 

4  l,r,.-.i 

— i.2(;o 

40,0:1c 

—0,04 

— 0.7t; 

r>i 

l.o:.s 

1,072 

1 ,  22!) 

o,o;;7 

o,c;i 

0.77 

fi2 

l,or.7 

1,002 

1,107 

00:17 

0,C2 

0.7  s 

M 

l,0"i.') 

1.712 

iicr. 

!         0,o;i7 

0,C0 

0.70 

r>4 

1.0,-.4 

1,7:11 

l,i:!:i 

1         o.o:;s 

o,r>!) 

O.SO 

.••5 

+  l,0.-)2 

-f  1  7r.o 

'    —1,100 

4o,o;is 

— o,r.7 

— O.Sl 

f>(l 

l.o.M 

1,7CS 

I,(m;7 

0,0:10 

o,ric 

0.S2 

f.7 

lOtO 

1,7S5 

l,o:i4 

o,o:!0 

o,r.4 

o,x:i 

f)H 

:       1.017 

1.H02 

1.002 

0,040 

o,.-i:i 

0,s4 

n;> 

M"i*i 

H   I,H17 

—O.ocs 

4  0,010 

— 0,.M 

— 0.s5 

THK   Oil  HIT   OF   UUA  XUS. 


1«» 


d>. 

6'/. 

('». 

rV 

y 

r9p 

9 

+  1.044 

6e 

edn 
—o.nb 

+0.041 

lie 

edn 

00° 

+l.«a:» 

—0.50 

—0.80 

til 

l.04:i 

1.H48 

o.ooi 

0.041 

0.40 

0.87 

(>i 

1.041 

l.s(12 

0.S07 

0.041 

0.47 

0.87 

f.:i 

1.040 

1.H7.') 

o.H;i2 

0.042 

0.4.'> 

0.88 

r,4 

l.o:is 

l.SSS 

0.70,S 

0.042 

0.44 

0.80 

«5 

+  l.o:iii 

+  1.001 

_o.7(i;j 

+  0.04:1 

—0.4  J 

—0.00 

«u-, 

1.0 :» 

1.012 

0.72H 

0.04:1 

0.41 

0.01 

fiT 

1  ().;:t 

1.022 

0.(103 

0.04:1 

0.30 

0.01 

«■< 

l.O.il 

l.o:i2 

0.(150 

0.041 

0.37 

0.02 

(1» 

l.o;to 

1.012 

0.(124 

0.044 

o.:to 

0.0:1 

70 

+  I.02S 

+  l.0".0 

— 0.5SS 

+  0.041 

—0.34 

— (t.04 

•       71 

1.(127 

l.OiV.t 

0.55:1 

0.041 

o.:t3 

0. 05 

72 

1  o2r> 

i.0(;7 

0.518 

0.045 

0  31 

0.05 

711 

1.0  2;  J 

1.074 

0.48:1 

0.015 

0.20 

0.0(1 

74 

1  1121 

I.OSO 

0.447 

0.0  »5 

0.28 

0.00 

7'> 

+  1.010 

+  i.osr> 

—0.412 

+0  1  15 

— 0.20 

—0.07 

70 

l.ois 

1.000 

o.:i7(i 

U.(>4(i 

0.24 

0.07 

77 

l.oir, 

1.001 

o.:ui 

0.04(1 

0.22 

0.07 

7S 

l.Olt 

1.007 

o.:j(t5 

0.04(1 

0.21 

0.08 

7'.t 

l.oi:{ 

2.000 

0.2(10 

0.04(> 

0. 1 0 

0.08 

MO 

+  1.011 

+2.oo;t 

_0.2:!:t 

+0.()4C) 

—0.17 

—0.08 

Hi 

I.OIO 

2.005 

0.1'  , 

0  040 

0.10 

0.00 

H-i 

l.oos 

2.007 

o.l(i:i 

0.04(1 

0.14 

0.00 

8;i 

l.ooc, 

2.007 

0.128 

0.((47 

0.12 

0.00 

84 

1.00.") 

2  (»0(1 

o.oo.t 

0.047 

0.11 

0.00 

«:> 

+  l.oo:{ 

+  2.005 

— 0.057 

+0.047 

—0.00 

—  1.00 

Sli 

1.002 

2.(101 

—((.022 

0.047 

((.07 

1.00 

H7 

1    IMM) 

2.(M(-j 

+ii.oi:i 

0.017 

0.05 

1.00 

8S 

O.It'.tS 

2.000 

0.048 

0.047 

o.o:> 

1.00 

m 

0.007 

1.007 

0  082 

0.047 

— 0.02 

1.00 

Jto 

+  0.00.") 

+  1.00:1 

+0.117 

+0.((47 

0.00 

—  1.00 

01 

O.'.IO.! 

l.oso 

0.152 

0.017 

-f  (1.02 

1.00 

!t-2 

0.002 

l.osi 

0.1 8(! 

0.017 

0.03 

1.00 

',t:t 

0.000 

1.07S 

0.220 

0.047 

0.05 

1.00 

04 

o.oss 

1.0-2 

0.254 

((  01  , 

0.07 

1.00 

05 

fO.OHT 

+  1.0(15 

+  0.287 

+0.((47 

+0.00 

—  1.00 

Oil 

0.  OS,-) 

1.!I5S 

o.:i2l 

0.047 

0.11 

0.00 

07 

(t.OS» 

1.050 

0.;i54 

0.047 

0.12 

0.00 

OS 

O.OSO 

I.oi:! 

o.:i87 

0.04(; 

0.14 

0.00 

00 

o.oso 

1 .  o.i;{ 

0.421 

0.04(1 

0.  HI 

0.00 

lIMI 

+0.070 

+  1.021 

+  0.45:1 

+  0.040 

+  0.17      ■ 

— 0.08 

101 

0.077 

1.014 

0.487 

0.040 

O.IO 

0  08 

1112 

(t.07« 

l.OOli 

0.510 

0.040 

0.21 

O.08 

lo:! 

0.074 

l.sit:{ 

0.551 

0.040 

0.22 

0.07 

lot 

0.072 

l.ss-2 

0.582 

0.040 

0.24 

0.07 

10.") 

+0.071 

+  l.S(iO 

+  0.(114 

+0.045 

-fd  20 

—0.07 

10l> 

o.oc.o 

I.S57 

0.(145 

0.045 

0.28 

0.00 

107 

o.oi;s 

is  (4 

0.(177 

0.015 

0.20 

0.00 

los 

o.ot;7 

>.S20 

0.707 

0.045 

0.31 

O.05 

loo 

O.Oli,") 

1.S15 

0.7:!7 

0.014 

0.33 

0.05 

no 

+o.oi;4 

+  1.S(I0 

+  0.708 

+0.014 

+  0.34 

—0.04 

111 

0.0112 

i.7sr, 

0.708 

0.044 

0.30 

0.03 

112 

0.0(11 

1.770 

0.827 

0.044 

0.37 

0.02 

U.i 

0.0.")0 

1.754 

0.855 

0.04:1 

0.30 

0.01 

114 

o.o:>s 

l.7;is 

0.SS4 

0.043 

0.41 

0.01 

lir. 

0.  o.")(; 

+  1.721 

+  o.oi:{ 

+  0.043 

+0.42 

— 0.00 

11  c 

0.0;")") 

1.704 

0.042 

0.042 

0.44 

0,80 

117 

0.0.")4 

l.lisii 

0.070 

0.042 

0.45 

0.S8 

UH 

o.o.w 

l.ddH 

0.007 

(1.041 

0.47 

0>7 

no 

+0.0;")  1 

+  1.(150 

+  1.025 

-\  11.041 

+  0.40 

_-0.h7 

^:a? 


m 
li 


T 


Itil 


T  II  K   O  U  B  I  T  OF  UK  A  N  US. 


d>. 

d-K 

&/. 

'■'(' 

''(' 

«5|' 

a 

0. 

de 

edn 

d( 

ve 

e-/n 

{■Mt" 

-fO.lC.O 

+  l.r.:il 

+  l.o:.i 

+  0  oil 

+0.f)0 

— o.sn 

121 

0.1»4!» 

l.i'.ll 

1         1.077 

0.040 

0.:-,l 

o.sri 

\.i-2 

0.!»H 

1..V.IJ 

l.lol 

0.040 

0  :);l 

0.M4 

IJ3 

0  !t47 

l.ii71 

1.1211 

O.O.'I'.I 

0..',4 

O..H.J 

124 

O.it.) 

i.:..-.i 

1.1. -.4 

o.o:i<.) 

u.r)i; 

0.^*2 

I3.i 

-f0.'.t44 

t  l.'i.lO 

+  1 . 1  ><0 

+o.o;iH 

+».:)7 

—O.Ml 

I2*i 

O.DI.'t 

l..')0<.l 

1.20.-) 

o.o:is 

\).M 

{).m 

127 

0.1I42 

I.4SS 

I.22'.t 

0.0.17 

u.r.o 

0.7'J 

I2S 

O.'.ltl 

l.4i:ii 

1         I.2...I 

0.0.17 

U  Wi 

0.78 

1 1't) 

O.'.MO 

l.44:i 

\         1.277 

o.o:i7 

u.f-.a 

0.77 

l.'IU 

-fl».'.t:il» 

+  1.421 

+  1..IU0 

+  0.0.111 

+  0.04 

— 0.7li 

III 

0.'.t,l7 

I.:i!i7 

1  .122 

o.o;in 

O.lili 

0.7.^ 

l:iJ 

O.'.KKi 

l.:i74 

1  :IU 

0  o:ir) 

0.07 

0.74 

l;t;j 

O.'.KIII 

1   .l.-iO 

l.:u;7 

o.o;t4 

O.II.S 

0.72 

III 

0.  !i;i.") 

l.;i27 

l..|M>( 

O.O.'II 

0.70 

0.71 

i.i;. 

-|-o.!i:i4 

-|-I.;i02 

+  l.lll!» 

+  o.o;!;i 

-|0.71 

—0.71 

I;ir. 

O.K.I.I 

1  277 

l.4:;it 

o.o;i,'i 

0  71 

0.70 

l:i7 

0.'J.12 

1  2.-.;i 

I.4.-.I 

U.ii:i2 

0  72 

o.os 

l.is 

(>.!t.ll 

1.22S 

1  470 

0  O.'ll 

0.74 

0.07 

111) 

O.'.I.'IO 

1.202 

I.ISO 

O.O.'il 

0.7') 

oil) 

110 

+lt.!l2!l 

f  1.177 

+  i.:)Os 

+  0.0.I0 

+  0.7ii 

— 0.04 

141 

o.'.tis 

l.l'il 

l..'.27 

u.(i:io 

0.77 

0.0:1 

142 

0.'.I27 

1.124 

l..^4.') 

0.(120 

0  7S 

o.r>2 

lt:i 

0.!I27 

l.lltM) 

l..'>r>2 

0.(I2S 

0.7!» 

0.110        1 

111 

O.llL'li 

1  1172 

l.'iXO 

0.02S 

O.NO 

0..'.!) 

11.) 

)-il.'.l2.') 

+  1  1114 

+  I..-.1H; 

+  11.1127 

-t  O.Ml 

— o.;,7 

Ill-, 

O.'.t-.'l 

1   Olli 

1  i;i:i 

0.ll2l-| 

(I.X2 

o.m; 

117 

0.!ij:| 

O.'.N!) 

1  i;2!» 

o.ii2r> 

o.s:i 

0..'.4 

US 

o.:ij2 

o.!ii;2 

i.iiii 

0.02.'') 

O.Ml 

o.r>;i 

ll'.l 

o.;iL'2 

(l.'.l.ll 

l.r..-.!i 

0.II2I 

0..S5 

u..->i 

ir>o 

f(l.li2l 

fo.ooc, 

+  1.1174 

+0.02:1 

+o.,s(; 

—o.'ll) 

i:.l 

0.'.t21 

0.S7S 

l.i-iss 

0  l»2:l 

0.M7 

0.4!l 

ir>i 

o.;i2o 

0..s4'.t 

1  702 

0.022 

0.M7 

0.47 

Ia3 

0.!M!» 

0.S20 

1.714 

0.021 

0.,H.S 

0.4:1 

l.'.4 

oil!'.) 

0.7!»2 

1.727 

0.021 

0.M9 

0.44 

l.i5 

-|-0.'.ll<» 

f0.7ii2 

+  1  710 

-H»  020 

f  O.'IO 

—0.42 

I'llt 

0.!IH 

o.7;i;j 

1.7;-.l 

O.OI'.I 

O.'.H 

0.41 

1.^7 

O.'JIS 

0.704 

i.7ii;i 

o.ols 

0.111 

0.:!!) 

l.->s 

o.'.tn 

o.i;7l 

1  77.1 

O.OIS 

0.<I2 

o.:t7 

i:i!» 

0  !l|l> 

O.lll.') 

l.7s.t 

0.017 

O.li.l 

o.:n; 

I  no 

+ii.:ilil 

-f-ll.lil.') 

+  l.7'.i:i 

+  0.011; 

+  0.'.i4 

— o.:ri 

1(11 

II  '.(!.■) 

o..">s-i 

1.  HI  I.I 

dill.') 

O.'.l;') 

0.:i.. 

lr,2 

(I.'ll.j 

0  ."..").•) 

I.Hll 

O.lll.') 

O.il.'i 

o.::i 

n;:{ 

((.  ;»).■) 

0  .'12.') 

l>20 

0.III4 

O.'.lll 

0.2!l 

iiii 

(t.'.ii.-i 

II.  C.ll 

I.S20 

0.01:1 

O.tlll 

0.2s 

Id,') 

f-O.'.Ill 

-+11.41;.-) 

+  l.s:;i; 

\  0.012 

+  0.'.I7 

— o.2ii 

Dii; 

(l.!)ll 

(1.  (:!.•( 

l.Ni;i 

O.OII 

0  !I7 

0.2» 

Iti7 

II. '.Ij  1 

II  lll.'l 

I.Mll 

O.llll 

0.07 

0.22 

ic.s 

O.'.lj.l 

o.:!7:i 

I.S-,;-) 

OHIO 

O.'.IS 

0.21 

KiO 

It.  11  I.I 

0  :!»:t 

I.HI'mI 

O.OO'I 

0  '.H 

0.1 '.I 

170 

+o'.i|:i 

+o.:ill 

+  l.sr,-, 

+  O.I)O.S 

+  (I.!IS 

—0.17 

171 

0  '112 

0.2><0 

1    H70 

0.007 

O.'I'.I 

O.ill 

172 

o.;ii2 

0  24!) 

l.-<7.^ 

o.iior. 

0.!l!l 

0  14 

I7.J 

0.!II2 

(l.2l!l 

I.S70 

o.ooi; 

0.!l!) 

0  12 

174 

0.!I12 

0.1H7 

I.HM2 

0.00') 

0.!I!1 

on 

175 

+  0.1III 

+0.1. "ill 

+  I.HKt 

+  0.004 

+  1.00 

—0.0ft 

170 

O.'.H  1 

0. 1211 

l.sstl 

0.00:1 

1.00 

0.07 

177 

o.'.tn 

0.0!I4 

I.HS.H 

0.002 

1.00 

O.ll'i 

17S 

0  '.III 

o.oi;:i 

I   .SS!» 

0.002 

1.00 

o.o:t 

I7!» 

+  0.'.l|| 

}o.o:il 

+  I.HOO 

+  0.001 

f  1.00 

—0.02 

Till-:   OK  HIT   OF    UK  AN  US. 


160 


In  the  oqiiutions  of  coiulitioii  tea  yenrs  lias  bivn  ndoptrd  for  the  unit  of  time, 

ill  onlcr  to  iiiuki-  tlic  gcMunil  valiii.>  of  tlio  cuctKciciitH  uh  iii'iiily  iqinil  as  possihlr, 

iiiul  llic  lime  liiis  hi'cii  (duiitid  from  tlio  cjiorli  1N;J0,(),  in  order  to  liave  tlie  jiosi- 

tive  and   negative  values  of  t  in  tlie  efinutioiis  mon;  nearly  i»aluiired.     To  distin-- 

giilHli  theHu  values  of  lie  and  iii  they  aru  marked  with  an  accent.     'I'iiis  unit  of 

•) 
time  gives  (),HJ)1 1  for  the  value  of  '   in  arc,  wliencu 


('li 


'I'lie  e«|iiations  of  condition  are  now  formed  by  putting  in  the  preceding  equa- 
tions for  heliocentric  longitude  and  ratlins  vector 


6X 


67. 


f'?.  ,       .      {jT. 


fu, 


h-K = Th  + :  fit-  + . " ,v  +  T e\n  +  : , V 


vv 


oe 


t:a.1 


O'i 


•        (Je  I'H  ve        '   «<',T 


1^ 
I'or  the  coefficients      "  have   beiii  taken  one-hiiiidr<'dtli   the  iierturbatioiis  of 
da'  ' 

loiif,'itu(l(!  jModiucd  by  Neptune,  as  given  in  the  heliocentric  ephunuris  at  the  end 

of  C'lia[it<r  \'.     'I'he  corrected  muss  of  Neptune  will  tlitn  be 

Finally,  1  n-niark  that  all  the  preceding,'  coniparistms  are  made  with  the  helio- 
centric epheiiicris  as  printed,  without  tiie  correction  indicated  in  the  colnniu 
adjoining  it,  but  in  the  following  eciuations  this  correction  is  for  the  first  timo 
introduced. 


Kijualioiin  I  if  fduilltioii  yinii  hi/  the  ('ornrti:,)is  in  Loiiyitndr, 


I 

o.(».),v 

—  0.70' «' 

— O.IO.V 

+0.0:U7t 

+0.1 '2W 

=  +  I'M 

••> 

0.10 

—  1.11 

-t-o.oi 

—0.10 

+0.1-2 

=  +  '2  .2 

a 

o.;ji 

-  '>..>;» 

-|-0.:{:{ 

-f  ().")! 

-0.-2-2 

=  +12.8 

4 

0.3-2 

—  'i.'t'i 

4-0.  •27 

-f0.">8 

—0.18 

=  +11.8 

T) 

(».;«) 

—  2.:Ji 

+0.1-2 

+o.r)0 

—  0.00 

=  +ll.() 

(1 

().1!> 

—   1.40 

—0.01 

-fO.:{0 

0.00 

=  +  5.0 

7 

0.1!) 

-   1.18 

— O.'.'i) 

+o.-,';j 

+  0.-2-2 

=  +4.8 

8 

1.41 

—  ().7rj 

— '.'.(iO 

—0.7") 

+-2.47 

=  +28 

y 

l.OO 

—  ;■).()  1 

—  1.08 

—0.7-2 

+  1.8") 

=  +   1.1 

i(» 

1.07 

—  4.01 

-1.04 

— o.so 

+  1.8.7 

=  +1.7 

11 

l.Oi) 

—  4.S7 

—  l.S,") 

—  1.04 

+  1.8.") 

=  —  0  .") 

li 

o.-;j 

—  ;{.()() 

—  1.04 

—0.07 

+  1.10 

=  +   1  .0 

V.i 

o.:)4 

—  '.».•.'() 

—0.71 

—0.78 

+0.82 

=  +1.4 

11 

().;m 

—  2.10 

— 0.08 

—0.81 

+  0.81 

=  —  0  .0 

1.-) 

0.7:J 

—   'IS{\ 

— O.S-2 

-l.Ki 

+  l.o:J 

=  —  0  .9 

i(> 

o.:n 

-  1  ;{!) 

— o.;u; 

— o.(;o 

+0.40 

=  -  0  .2 

n 

0.."»() 

—   '.'.O") 

—0.47 

— 0.0"» 

-f0.70 

=  —  0  .4 

IS 

o.:n 

—  l.:5:J 

— O.'JO 

—0.0") 

+0.44 

=  +  0  .0 

1!) 

0.7:) 

—  '2.M 

-0.:J8 

-l.:]4 

+  0.8-2 

=  -   1  .0 

!"1 


166 


T  il  E   O  K  H  1  T   O  V   U  R  A  N  U  S. 


20 

0.56  y 

—   1.89^, 

-0.22,V 

-1.04.S« 

+(V58^m' 

=  —  0".9 

21 

0.5H 

-    1.92 

-0.13 

-1.12 

+0.55  ' 

=  —  (» .5 

22 

0.;J8 

—   1.15 

+0.09 

—0.74 

+  0.-24 

^  -  0  .2 

23 

o.;}8 

-    1.11 

+0.15 

0.73 

+0.21 

=  -  0  .3 

24 

1.15 

-  3.18 

+0.65 

—2.13 

+0..")0 

=  +  0  .6 

25 

1.13 

-  2.81 

+  1.!1 

—  1.91 

+0.19 

=  +  0  .5 

26 

0.56 

-    1.33 

+0.62 

—0.89 

+0.05 

=  +0.1 

27 

1.12 

-  2.54 

+  1.38 

—  1.68 

0.00 

=  +  2.1 

2S 

0.56 

-   1.21 

+0.75 

—0.79 

—0.04 

=.-  0.1 

29 

0.70 

-   1..V) 

+  1.09 

—0.97 

—0.10 

=  +    1.6 

»0 

1.11 

—  2.19 

+  1.73 

-1.32 

-0.-23 

=-+  2.2 

31 

1.09 

-  2.04 

+  1.81 

-1  14 

—0.-29 

=  +   1  .3 

32 

0.5(5 

—  0.!>8 

-H»-06 

-0.51 

—0.18 

=  +  0  .7 

33 

0.55 

—  0.!>1 

-I  i.OO 

—0.43 

—0.21 

-+    1.1 

34 

1.08 

-    1  CO 

+  2.0'< 

-0.71 

—0.46 

=  +   1.5 

33 

l.(i:< 

-   2.37 

+  3.11 

—0.8-2 

-0.74 

=  +  2 .(» 

35' 

1.07 

-    1.45 

+  2.08 

—0.38 

—0.52 

=  +  0  .4 

35* 

i.oe 

—   1.33 

+2.10 

-  0.-22 

—0.55 

=  +  0  .8 

30 

1.58 

-   1.S3 

+  3.18 

-  O.tiO 

— 0.S5 

=  +   0.3 

3-J 

1.05 

-   1.11 

+  2.10 

+0.08 

—0.59 

=  —  0  .8 

38 

1.04 

—  0.99 

+  '>.07 

+0.-2(} 

—  0.59 

=  —   1  .3 

39 

0.69 

—  0.58 

+  l.3() 

+0.27 

—0.39 

=z  +  0  .8 

40 

0.69 

-  0.51 

+  1.33 

+0.38 

—0.39 

=  +   0  .8 

41 

(».<)S 

-  0.44 

+  1.29 

-(  0.47 

-0.39 

0  .0 

42 

l.Ol 

-  ().,")6 

+  J.88 

+(t.si 

—0.59 

.4    0.3 

4:) 

0.67 

-    (».30 

+  1.20 

+  0.64 

—0.38 

—.  —  0  .5 

44 

1.00 

—  0.35 

41.72 

)  !.09 

-0.57 

=^4-0  .2 

45 

1.95 

—  0..")0 

+3.21 

4  2,39 

-1.12 

—  —  2  .5 

46 

2.42 

-  (>.:n 

+3.77 

+3.21 

-1.3(i 

=--  —  2  .() 

47 

2.42 

-  0  12 

+3  ,-,3 

+  3.52 

-1.37 

=^—  2.4 

48 

2.39 

+  0.12 

+  3     i 

+3.(i9 

—  1.33 

=  -  4  .9 

49 

1.91 

-I-  v».28 

^2.;i7 

4  3.14 

-1.07 

rrr    _    0.1 

50 

1.91 

;-  0.48 

+  2.15 

+3.30 

—  1.08 

=  —  2  .4 

51 

2.37 

+   0.S3 

+2.31 

4  !  32 

-1.34 

=  —  4  .5 

52 

2.3(J 

+    J. 06 

+  2.02 

44  u 

—  1.36 

=  -  4  .3 

53 

2.37 

+   1.32 

+  1.68 

44.62 

-l.:57 

■=.—  5.5 

54 

2.85 

+    l.S(> 

+  1.66 

45.(11 

-I. (IS 

=  —  7  .5 

55 

2,84 

-f  2.17 

+  1.22 

-]  5.7  1 

-1.71 

=  -  4.1 

5(J 

2.84 

f-  2.44 

+  0  81 

45.84 

-1.74 

=  —  2.1 

57 

2.8? 

+  2.71 

+  0.K» 

45  8H 

-1.78 

=  -   1  .7 

5S 

2.83 

+  3.0() 

— 0.()(» 

4  5..SS 

-^  1.S2 

=  -   2  .0 

59 

2.82 

+  3.28 

-0.41 

45.S4 

—  l.s(; 

:--    f    0 .3 

60 

2.83 

+   3.57 

— (».N1 

45„si 

-  l.iU 

=-  +   1  .2 

61 

2.85 

+   3.00 

-1.27 

4  5.78 

—  1.98 

-  +   4  .4 

62 

2.85 

+  4.21 

-1.74 

-f  5.(il 

-•2.02 

==  +   4  .M 

63 

1.91 

+  3.02 

-1.42 

43,(i9 

—  1 .39 

=  +  4  .0 

64 

2.85 

+   -l.-M 

-2.50 

45.:;2 

-•2.11 

=  +  6.0 

65 

1.91 

4-  3.3!» 

—  l.'tV' 

4  3.J(i 

^1.17 

=  +  5.6 

66 

1.92 

-f  3.61 

-2.18 

43.3-2 

-1.52 

=  +  4  .3 

67 

1.93 

+  3.8-2 

-2.40 

+  3.17 

l..-)6 

=3  4    () .() 

68 

1.94 

+   4.05 

—  2.65 

4  2.0s 

-    1.(10 

.-:-       f        7.3 

69 

1.94 

+  4.20 

—  2.  S3 

+2.80 

-1.61 

==  -(    (i  .5 

k^  ■'«--: 


THE   Oil  BIT   OF   URANUS. 


1()7 


70 

1.}).-.,V 

-t- 

4.44^h' 

-3.()4'e 

-f2.59c,^n 

—  l.(58,'iH 

=  +  (5".8 

71 

l.»8 

-h 

4.72 

-3.23 

+2.37 

-1.73  ' 

=  +    8.1 

72 

i.})y 

+ 

4.93 

-3.44 

+  2.15 

-1.7(5 

=  +9.1 

7:J 

1.99 

+ 

5.13 

—3.58 

+  1.91 

-1.78 

=  +8.7 

74 

•i.oo 

+ 

5.35 

—3.73 

+  1.(54 

—  1.80 

=  +  8.1 

7.) 

'2.M 

+ 

7.02 

-4.N3 

4  1.73 

—2.30 

=  +10.(5 

7(J 

2.54 

+ 

7.34 

—4.9(5 

+  1.33 

—2.29 

=  +10.9 

77 

2.5(} 

4- 

7.(54 

-5.97 

+0.97 

—2.31 

=  +  8  .2 

7s 

2.»»7 

+ 

(5.38 

-4.14 

+0.47 

—  1.8(5 

=  +  (5.3 

7i» 

2..">!) 

+ 

S.25 

—5.20 

+0.21 

—2.32 

=  +  7  .0 

m 

2.()(> 

+ 

H.51 

-5.21 

-0.15 

—2.29 

=  +  (5  •<> 

81 

2.11 

+ 

7.U 

-4.19 

-0.51 

-1.N3 

=  +  4  .3 

S'> 

l.fiO 

+ 

5.5(i 

-3.13 

— 0.5M 

-1.3(5 

=  +  3  .9 

Kl 

'Hil 

+ 

9.59 

-5.12 

—  1.37 

—2.20 

--+    1.1 

84 

2.G8 

+ 

9.!>2 

—  5.00 

—  1.80 

-2.14 

=  -    1  .s 

85 

2.17 

4- 

8.22 

—3.92 

-1.75 

-1.(57 

=  —   3  .8 

m 

2.73 

+ 

h).()l 

-4.97 

—  2.59 

-2.01 

=  —   !).0 

87 

2.20 

+ 

8.78 

— 3.(i;j 

—2.33 

—  1.52 

=  —  9  .5 

88 

l.(>5 

+ 

(5.74 

—  2  58 

-  1.9(5 

—  1.08 

=  -10.5 

8!) 

2.77 

+ 

11.(51 

—  4.04 

-3.(51 

-1.(59 

=  —  19  .9 

2S3.(54,V 

+   41l.3(i,^u' 

— 151.()3^; 

414.3(5 

+  1(519.14 

— (5N9.ll 

151.(53 

—  (SsD.ll 

+  557, S2 

247.23 

+   2(!0.2(5 

+  3S.»5 

ri(5.0:{ 

-     43(;.(t2 

+  122.S8 

'I'lic  lollowiiifj  arc  tlu'  iipproximiifc  normals  to  which  these  r()!iatioiis  fii\e  rise. 
Inaccuracies  lieiii;,'  delected  ill  .^cMTal  of  the  e()liatiolls  ot"  conditinil  alter  tiiese 
normals  were  formed,  they  <lo  not  accurately  corr<'sjioiid  to  those  ((piatious  as  written. 

+  247.2:kA7i  —  17(5.03,W  =-     (   123'.5 

+  2(50.2(5  —43(5.02  ^.    \    103.2 

-f    3S.45  +122.8S  =— :t99.M 

-I  (lis.  I.-,  —19 (.(50  =  -j  2(57  .3 

-  191. (50  -|l(i{.13  :.:.—  12S.1 

The  values  of  tile  u!i!>uov,-n  ([iniitities  deduced  frc.n  tlicse  ufU'iiials  were  snlisti- 
tnted  ill  tiie  <i|natioiis  of  conditioti,  and  a  fartiier  approxi. nation  was  made  iiy 
sohini:  tlie  eipi  ifioiis  ;j;i\('n  liy  tin-  residuals.  The  followiii:,'  are  the  tirst  approxi- 
niatioas  ^iveii  liy  tiie  luMinals,  and  tlie  tinaily  roncluded  ccnctions 

I'rcliiiiiiiiiry.  h'iiial. 

— 15.00  '  — l:i.U 
_  o.;V2  _  o.:U5 
_    I.J.-,         _  4.04 

10,Vn  =  f^/(',—    I."i3         —    1.33 
V,  —   3, IS         -  3.44 

Mass  of  Neptune  ,^l,^  ^,,,<,^. 

The  final  \alues  of  the  corrections  Ix'in;^  snhstituted  ir  the  i'(pi!itions  Ieav(<  the 
following;  s\st<'m  of  residuals,  or  outstanding  escesses  of  tiie  oliser\ed  loiif^itudes 
over  theory.  Colunni /'(V  f^i\es  the  residual  of  the  e((Uatioii  itself;  the  prohahle 
error  of  which  lias  always  heeii  jniliied  to  he  0."">,  whih  in  c<'!uini'.  ?!  this  residu.d 
is  dividetl  hy/'to  oltain  the  roidiial  correction  cd' the  loii:,'itii(ie  itself  'the  \alues 
of  the  factor* / are  found  with  tin  original  (upuitious  on  pa),'es  159  and  1(50- 


Cj?«» 


168 


THE   OKIIIT   OF    URAXUS. 


.  of  Eq.    Year. 

f)l 

1       1691.0 

-0.5 

-10. 

2      1T1"».'2 

—O.S 

—  10. 

;{      IHS.H 
I      n'lO.H 
T)      17  •»:{.}) 
(i      17,6.7 

+1.5 

-f  O.M 

— o.-i 

+  ().() 
-    1.0 

7  1 7(;!).0 

H  1 7S-.>.0 

))  ns;t.() 

10  ITSI.O 

11  nsvo 


12 

i;j 
u 
i:. 

Ki 

17 
IS 
1!> 
20 
21 


;^2 
;j.v 


1 7SS.0 
1  7Si).0 
175)0.0 

nin.o 
n!»2.o 

17!):l.O 
17!M.O 
17!).").0 
1706.0 
17!»7.1 


22  1S()0.2 

2:{  I  MO  1,2 

24  I  s( )•,'.:{ 

2.")  lso.V;J 

2(1  ls(Ki.:j 

27  ls07.;{ 

2H  1M0H.:{ 

20  lson.:J 

:J0  isio.:} 

;Jl  isll,;} 


1  SI  2.4 
isl:).4 
IMI.I 

IHI.VI 

l^K!  t 


—  0.2 

-fO.l 
— O.j 
+0.(i 

—  1.0 

+  1.2 
■fl.7 
— (t.5 
-0.7 
0.0 

-0.1 

-0.6 
— 0.") 

~o.(; 

-1.4 

—  1.4 
—0.6 

-1.4 

-1.0 

O.l 

1.0 

-f-0.1 

-0.1 

—<».!) 

-0.;) 

-fO.l 

—0.4 

o.:» 

-  (».H 


—  1.0 

+  0.2 

—  0.;> 
+  0.6 

—  1.0 

+    l.S 
+  U.4 

—  1.0 

-  1.0 
0.0 

0.2 

-I     I  •■■» 
-   0.i» 

-  1.0 

-  1.2 

—  4.2 

-  4.2 

—  0.6 

—  1.4 

-  2.0 

-  O.l 

-  2.0 

+  0.2 

-  0.1 

-  Us* 

—  o.rt 

•f  0.2 

-  0.4 

—  0.2 

—  «.H 


So.  of  K(|.    Ycnr. 


;n 

;js 
;i!) 


45 

46 
47 
4S 
4!) 

50 
51 
52 


1S17.4 
ISIS.4 
1SI!).4 
1.S20.5 
1N21.5 


40  1S22.5 

41  l,sv';j.5 

42  1S'.'4.5 
4:{  1,S25.5 
44  1S26.5 


1S27.7 
1S2S.7 
is-,';).  7 
1s:M»,7 
|s:tl.7 

ls:t2,7 
lN:!:t,H 


ls:n,M 

5;{  ls:i.-,..s 

54  l.s;{(i.S 

55  ls:n.S 

56  ls:{,s.H 

57  ls:i!).S 
5.S  l,s|(),S 
5!)  ls||,S 

60  1S4'.'.,S 

61  lH4:J.s 
(»2  1S44,!» 
M  IM.V!) 
(it  1sk;,!» 

65  lsl7,i) 

M  ls|slf 

67  lH4i).i) 

6s  1M.V),<> 

6!)  1M51.0 

70  Is.V.'.O 

71  ls.-,;l.i) 


'3? 

-0.;} 

—0.7 
-1.4 
—  1.4 

+0.0 

+  1.4 

+  0.6 

+0.7 
+2.0 

+  1.0 

+  2.5 
+  2.S 
-f  0.4 
+  41 

+  0  5 
+0.1 
—  1.4 
-:1.4 

—0.6 

+0.5 
-0.4 
-1.7 
— 0.:j 

—0.5 

+ 1 .6 

+0.7 

i  o.(; 

-0,4 

+  0.6 

l.() 

+0.2 

|-o.:j 

-1.2 

-  1.2 
-0.2 


fit 

n 

—  ().;) 

—  0.5 

—  1.4 

—  1.4 
+  1.4 

+  2.1 

+  0.0 

+  1.2 

+  1.0 

+  2.0 

+  0.5 

+  1.0 

+  1.1 

f  0,2 

+  2.0 


+ 


0.!) 
0.2 
0.0 
0.6 
1.1 


—  0.2 
+  0.2 

—  0.1 

—  0.6 

—  0.1 

—  0.2 

+  0.5 
+  0.2 
+  ().:} 

—  o.l 

+  o.:i 

—  o.s 

+  0.1 
+  0.2 

—  0.6 

—  0.6 

—  O.l 


THE   O  U  m  T   OF   U  K  A  N  U  S. 


169 


Ni).  Ill'  Ki].    Yi'iir. 


NO     i.sC);;.o 


ni 


H 


Vi 

IH.-)!.)) 

+  0.5 

+  0.2 

r.\ 

IH.V).!) 

0.0 

00 

14 

isr.G.'J 

—()..■) 

—  o.-J 

1") 

is.ys.O 

— O.'i 

~    0.1 

It; 

l.s.-,!).() 

1  1).!) 

-f  o.i 

^  •» 

1 1 

Ihdd.O 

- 1.1 

—  o.i 

"S 

18(U.(> 

— 0.(> 

—  (),:{ 

1!) 

iw;-,>.() 

— o.(; 

-    O.'J 

40.  v'  -\-     0.1 


No.  of  V.i\.    Yi-'iir. 


Pl 


u 


HI 

is(;-i.o 

+0.7 

+ 

0.4 

S-2 

isd.j.o 

+  2.-i 

+ 

i.r. 

s:{ 

1,S(!().0 

+  1.0 

-f- 

0.4 

Nt 

1M)7.0 

+  ()..-> 

-t- 

o.'i 

8.5 

INIJS.O 

+  0.4 

•1- 

0.'v» 

Hi 

l.S(i!).0 

-l.iJ 

— 

O.H 

NT 

IHIO.O 

-o.l 

0.0 

HM 

1.S71.0 

-l.'i 

— 

o.s 

W) 

187 '2. 1 

+0.4 

+ 

o.'i 

.\  simple  •(limcc  at  tlic  roiirsc  of  tlio  rcsiilimls  shows  (1)  lliat  tlnir  i>rol»iiliIc 
visliic  i.s  fonsidcialiiy  !j;i(at<i-  tliaii  the  (inilmltlu  error  attril>nti'<l  to  tli  •  ciiuafioiis 
ot'  /.oiuiitinii,  hciii},'  more  marly  0".7  than  WS^y  ami  yet  larger  in  tlic  later  y<-ars ; 
('.')  tliat  diiriii','  rertaiii  periods  they  are  of  a  systematic  eiiaracier.  J).irin<,'  llie 
years  I74H  to  \'>'t'-\  tlw  oliservatioiis  show  a  de<ided  positive  eorreeti'tii  to  tiie 
theory  of  a  ma;,'iiif iide  },'reater  than  we  can  consider  prol»al)le,  anionntin;,'  to  aliont 
one. third  of  a  second  of  time  in  the  mean  of  Ihadley's  two  ol)servation8  of  1748 
and  n.'»:).  Ahont  ISOO  (he  correction  hecome.s  ne<,'ative,  and  so  continues  for 
'.'0  years  with  an  averaj,'e  v;dne  <if  ahont  I".  In  l^.M  it  suddenly  hecoim  s 
positive,  and  so  continues  until  l8;j;l.  From  this  year  forward  the  residuals  are 
not  systematic  in  character. 

In  order  to  sliow  clearly  the  general  coursr'  of  the  out-itandin<,'  corrections,  they 
hiiM'  h'  in  <livided  into  groups,  generally  including  aliout  five  years  ciuh.  The 
mean  outstanding  concciion  for  each  group,  taken  with  respect  to  the  vvi  iglils 
iiidirnted  Ity  the  fa<tors  y",  is  as  follows.  In  the  colunui  i  is  shown  what  the 
proliahle  error  of  ihi-  residual  should  he  if  the  wi'ights  iissigiicd  to  the  several 
equations  were  strictly  correct,  ami  110  Kystematio  errors  wcr-  priMiit  either  in 
theorv  or  ohservatioii. 


$« 


Y.ar, 

m 

f 

n 

ni.V'i 

-^ 

Kt. 

+  <!. 

n.-)i.i 

i 

;i.7 

+0.M 

ntiK.o 

-- 

1  0 

i  •-'.  ■-.             ' 

ns;!.:^ 

O.IM 

+  <>.«':» 

I".!»0.0 

; 

0.(i'.» 

d  <»■«<» 

170.'i.O 

— 

O.."),") 

t  0.11 

ISO  •.>.(> 

— 

l.'j:. 

+  <t.l.") 

1S(  )!>,."» 

1.00 

+  (!.:il 

isio..-. 

o.:!7 

+0.:j'2 

IMI  .'. 

0.117 

-\  0.',':l 

islii,.-) 

o.:n 

XS^AH 

•Xi       M..y 

1873. 

Ynir. 


18'.>».8 

-I 

l.:,o 

+0.1  () 

ISJO.7 

+ 

0.01 

+0.10 

|8:r..'.> 

— 

0.'.>7 

+  0.00 

I8;i!).s 

— 

0.17 

+  <>.i>7 

184  4.8 

■1 

0.1  1 

+o.o:J 

isio.o 

— 

o.-Jl 

+0.11 

is.-,  1.0 

.... 

0.1  1 

+0.11 

Isdo.o 

— 

0.1  :J 

+  0.0!> 

ls(i.',.o 

■f 

().:{() 

+0.10 

ls-,0.0 

— 

O.'Jl 

ri  <».n 

M 

i 

'IF  = 


k 


no 


THE  ORBIT  OP  URANUS. 


|i^ 


A  Miiiplc  glance  at  the*  rcsitluiils  il  hIiowji  that  tlicy  nro  much  {greater  Ihuii  the 
purely  lu'cidi'iital  residuals  rehulting  iVoni  tlie  theory  of  least  sciuares.  We  may 
divide  th(>  putisible  cause!*  of  these  sytiteuiatic  enttrs  iutu  three  classes. 

1.  iSi/nlt iiiallc  h'rroiM  of  Olitnrnitwn. — These  may  result  from  deviation  of  the 
line  of  ct>llimatiou  of  the  iustruuuMit  from  u  true  great  cireh-,  or  from  any  i»ecu- 
liarify  of  th<?  observer  which  leads  to  his  registering  the  transit  of  I  rauns  ciirlier 
or  later  than  that  of  a  hsed  star.  If  we  conntare  the  corrections  lUrived  fiom  tin* 
work  of  ditterent  ohservafories  as  given  in  the  last  chapier,  we  shall  hud  fre<|nent 
cases  not  only  of  systematic  ditl'trences  lietween  <he  results  of  diti'erent  oitserva- 
tories,  hut  li(  tween  those  of  tlie  same  ohservatDry  in  two  siucessive  years.  An 
instance  which  particularly  attracted  my  attentiou  on  lirst  prepaiing  the  c(uii- 
ltaris(ms  of  theory  and  o!)servation  is  that  of  the  (ireei'wich  observations  for  ls;ii, 
which,  as  compared  with  observations  at  the  same  ol>seivntory  diiring  the  yem^ 
preceding  and  tnllowiiig,  seem  to  be  afieetid  witii  some  constiint  err.n  in  U.  .V.  of 
about  'i".  I  tind  that  this  disc  ppaney  can  be  attributed  only  to  the  original 
observations. 

'i.  JilrmiM  III  ihi  Tliriiin  i-nitiintii  il.-  ■\'\\i'><v  may  arise  tVotn  errors  in  the  jireceiling 
theoretical  computations,  fiom  the  omission  of  the  terms  of  th<'  second  otder  \my- 
(luccd  by  Neptune,  from  the  adoption  (d' an  erroneous  n\ass  of  Saturn.  u\-  from  tb<> 
attraction  of  an  nnknuwn  planet.  NN'ith  regard  to  the  probability  of  tin  >e  ditl'i  rent 
siinrees  of  ernii'  it  may  bi'  rem  iiKed  that  errors  of  C(im|iutation  seem  pnssibli'  imly 
in  the  terms  of  the  seennd  order,  that  the  mass  (d'  Satnru  is  taken  from  the 
exhaustive  disenssiun  of  the  Saturnian  sjstem  by  IJessel,  in  which  an  error  suffieient 
to  intluenee  tin'  theory  of  I  r.uins  si mss  higldy  improlmble,  atid  that  a  trans- 
Neptunian  planet  large  enuiigh  to  produce  a  sensible  deviation  of  the  orltit  of 
Iranus  fntm  an  ellipse  in  the  cotuNc  (da  eentnry  vvimld  be  too  large  to  have  es<'a|ie(l 
detection.  The  ehoii'e  of  tlie  elliptic  (liinenls(d'  I  raiins  and  N(  jitniie  is  snrh 
that  the  terms  of  the  serond  order,  due  to  the  a<tion  of  ^lcptune,  can  scare<  ly 
become  sensi'tli'  willun  a  eentnrv  of  tli<'  i  pm  h. 

U.  I'li'i'iirH  ill  t/ir  nirltiiiM  I'l  /iirllnim  l>i/  ir/iir/i  Tlnori/iinil  O/tsrrriitiiiH  iiri'  idih- 
jMiri'l.-~l\\  the  methoil  adopted  for  comparing  tbeorv  and  olis<r\ation  a  nnmber  of 
small  uncertainties  incident  to  the  iiii[ierfe:tions  of  the  older  data  id'  reduction 
iieci'ssarily  creep  in.  In  the  early  observations  tlie  imperfections  arise  prim  ip, illy 
from  the  niiceriiiint}  of  the  instrninenlal  correefioiis,  ,iiid  tlie  (ii>irs  in  the  adopted 
positions  of  llie  Inmlanniilal  stars,  and  indeed  in  nearly  ai!  the  ilata  of  ritha  (ion. 
In  tlie  lale  v  .irs  they  arise  principally  from  the  great  magtiitiwle  (tf  the  correction 
to  Ho.ivanl's  tables,  and  the  cotiseqnent  rapid  ( liange  of  the  corre<tion.H  to  the 
geocentric  epli>ii'  rides,  which  make  the  dcti  rminaJio!;  of  the  corrections  J^l  sumI 
N  from  theory  and  obsrr\ation  somewhat  niieertaiii.  l'".rrors  jVoni  ftii-<  source  will 
uecettsarilv  Iw  in  put  of  a  systematic  <  haracter.  and.  in  view  of  tie  :  possibilitv, 
I  re!.,'ret  not  bating  been  able  to  completely  re-r<(lue.-  all  tin-  observations  before 
INIO,  and  to  compare  all  since  dircctlv  with  ephennridt  s  eoinpnti'd  IVom  llw 
pr.nisional  theory.  In  ordir,  however,  to  test  fjie  ipiestion  whetlnr  tliej  are 
siMsilde,  I  leixi'  prepired  an  e|iloMneris  from  tlie  pro\isiftnal  theorv  for  the  three 
relent   oi'.positioiis  of   I'^ii!    2,  \h{,}  ;[.  aiid    In 7-.',  and  compind  it  dir.cily  with 


! 


Til  K    'I  U  MIT    UK    I'  UA  N  US. 


,171 


, 


till'  ulisrnaUniiN.  'I'hc  iMcfiu  <i>r)M'«-ti(iMs  ill  <;('occtifric  loii^^itiidc  for  •;i'(tiij>s  of 
(il»^<T\ati()iirt  ail'  ffi'.i'ii  in  <(iliiiiii»s  ("i),  riAuimi  (I)  sliuxviii}.';  ti;r  <  oriHTtioii  givi'V 
by  till!  work  of  tlie  last  i'haj)tii-. 


<)|i|msiti<jii      !S(.I   -i 


l^>:l    .! 


") 

(••i) 

+'r.« 

K/'.-r 

•i.!> 

•i .  K 

;f  ,0 

:i.l, 

2  » 

•.',1 , 

Mcjm   I -.Mi)     f'i.()i 


0) 

i;;> 

-1  '->".»> 

-r  •/'.«. 

•J .;{ 

2 ,:», 

t.'  1 

•,'  0, 

;{ .;> 

•>  .*». 

•J  .)i 

• » . « , 

-f  -J  .ti.'i 

-l-i.5! 

H7'i 

(1)  (•-') 

__H.;i  —  H.l, 

-7. a  -7.5, 

-11  -7.7, 

—7  .H  -7  .«, 

— 7  .(ij  —7  .82 


A  systciniitir  ilHI'i  r  iicc  of  0"  IC)  woiilii  >i  riii  to  lie  indiciiti'd,  iiiul  on  nrcomit  of  it 
II  con'cctiiid  iif  II", 10  was  iiiipl'i'd,  to  !ii>'  i»;iiij>aii,'^oii.'>  of  tlic  la.-l  frw  jrais  in 
I'liniiid^  titr  »'i|iiati!>iis  I'f  t'diiihtiiiii. 

Ill  uiu  of  tilt'  po.ssiliility  of  >yst(!iialir  i'rri)i<  fmni  tlii-.  .soun-c  it  may  he  coii- 
Kiili'icd  that  too  f^rcat  rclativi'  wii-^iii  !ia,s  In  t  u  ;!s,siirii(.(l  to  the  rtsult.s  of  ilic  later 
oljscrvatioii.H.  If  tin-  ii'siiliials  aiisf  froiu  i  rrois  of  coiiniarison  and  of  (Ihoij,  tliiir 
jirohahli'  Jiia;,'iiituili'  is  iwarly  as  j.'i-('at  ai  iuir  ipiu  Ii  a,>.  at  aiintlii-r.  It  may  llicrc- 
ton-  111-  niti'i'cstin;;  to  iiKin'.n'  what  r»'s<ik  wi-  vlnmld  |,'rt  it",  iiistriul  of  asNi;,'iiiiii{ 
Muh  ditt't'irnt  wci;;!!^  to  llir  niniiiaiiMHis  at  didVniit  ipoclis,  «c  M)n<,'li!  nnh  fur 
llic  l)i'st  jjrncral  a<.:iri".ii('i)l  willi  oliscrx.itiuns  iliniu;;  the  jicviud  ilir  jilaiii't  lias 
lici'M  ulisfTMil.  Till'  jKcri  (liiij;  >',  >ti'iii  of  Diraii  rrsidiiaU  will  riiahlc  ns  to  liisciiss 
tills  ([ncstion  i|iiiti'  ra'-ily.  In  llir  Hist  solution  wi'  sliall  rcji'i  t  the  rrsnlts  from 
I'lamsti'ml's  olwcrvations,  Mwinf^  to  tliiir  assurni  Mn< 'itaintj,  and  thosr  fmiu 
1,1'  Monnicr's  of  \'l\'.).  nwjn;;;  to  iIh'  ]iii<..ilil<'  nia!;!ilJ!i-!iii<iit  ol'  his  ijiiadnuit,  I'lic 
rijuatioiis  fioiii  ihi'  rciiiaiiiiiig  n  siilna'-,  uill  In-  \\u   (ollnwiiig: 


l.(),W 

—  7.()(V'm' 

4().s,w 

\\.le,Yn 

-o.a,vy 

■=r-..  4.:r.-i    w 

i 

I.I 

-•).0 

~  .'.0 

-  K.S 

4   1  S 

(►.IS 

I 

11 

-4.1 

l.» 

-  1  (i 

r  i.<; 

—  f  o.(i-,> 

I 

i.l 

-:!.!» 

-(>.() 

"•.'.0 

fi.'~' 

r--^-0.iM 

1 

):i 

-U 

-f-O.I.) 

-t>.l 

fO.o 

=  -1  .'.>;■) 

1 

M 

~'iA\ 

-Hl.;J 

-    I.M 

o<» 

=.  —  1    (10 

•I 

1  1 

-i:i 

-1  1.7 

1  ;i 

0  .» 

=  -  0  ;(7 

I 

1,1 

1,7 

*  'H) 

o.l 

tl.) 

=  -  0.:!: 

l 

1,0 

11 

ri\ 

(0.1 

-tl.O 

ss  -  0,:i7 

I 

1,1) 

0  ,v 

-)  l.H 

1  0,0 

0.(t 

ass  -•{  1  ,."iO 

•» 

10 

0.0 

1  1.4 

f  1.4 

-  0.:, 

—    iO.OI 

:) 

o;j 

ho..-, 

1  O.H 

4»M 

0.5 

=  -o.'j; 

:) 

(•.:» 

f  (>,!» 

^01 

,'<• 

0(i 

r^   -0     17 

:i 

!.(» 

1  1  ( 

— «ii 

-i  I.IJ 

-0.7 

=    1  O.H 

:» 

10 

!  1  '» 

-ft? 

f  Mi 

-  JI.M 

=:-    -0    •.>! 

:t 

10 

4*^.> 

1.7 

-fl.l 

(».«> 

=    on 

;j 

10 

fiU 

-  'i.O 

f  l>.  1 

0<l 

•«-^o  l;{ 

:t 

11 

,  :1.7 

-11 

-0.4 

0.11 

=  +0    Mi 

:i 

1.1 

-;  i.i 

*>40 

~l.i 

-OM 

=  —0  .;»l 

a 

17-2 


T  II  K   O  II  It  I  T   O  F    V  K  A  N  V  S. 


(living  tlios«>  uinofcon  cqtififioiis  «>i|iiiil  wci^'lits,  we  liavr  tliv  second  of  flio 
fi)lIowiii<,'  soliilions,  anil  tiic  second  »»f  the  series  (if  residuals  tiie  Hrst  eorres- 
pDiidinj,'  to  liie  |triinitive  Nolntion.  Solvin-;  liieni  a},'ain  and  assijrninf,'  tlie  wcijjlits 
attaehed  to  the  respective  e.inations,  wiiich  1  jndyo  to  he  inose  to  wliich  th<'y 
are  entitled  when  a  liberal  allowanec  is  made  for  systematic  errors  of  observa- 
tion an<l  of  comparison  of  the(iry  with  ohservations,  nddinj?  also  the  ecpiations 
given  l»y  the  observations  of  I'lanistead  and  he  Moiniier,  which  are  us  follows; 


^11 


111 


IC.iU)      0.0,W-O.Wn'-().U*e  4  O.OfS^Tt  4  ().UV=  -0".-,;  /,  .,i,^ ;  AVt,  1 
I'M) 


O.,'     -\.i       —(),;{     -|-().'.>        -jo.-i      ~  ^2  .'2        I 


we  have  the  second  solution,  and  the  third  series  of  ri'siduals. 

6*e,  0  -  (r.;!<) 

i*ni  0  —  (» .;w 

i*e  0  —(>.:};) 

efi^n  0  I  0  .'i.j 

Ay  0  _  1  .()-2 


m' 


Vear. 


1  a  0  r  il 


^J 


I  1)  »  7  II 
lU.SIIH  M.S. 


(-) 

-0.21 
-0.19 
-0.1 -) 
+0.1!) 
—0.1!) 


a; 


1 


ii;*)i.ii 

-10. 

^11. 

-I-.'. 

ni.vi 

-10. 

-     !). 

~  7. 

n:)i.i 

+  :U 

+   0.0 

+   'i.O 

IKiil.O 

-    1.0 

-  '.>.() 

—   1.!) 

ns:{.:{ 

—   O.IS 

—  o.:m 

-  0.17 

17!K).(» 

-f   o.d? 

(    <).!):{ 

+  O.H!) 

n!).V(» 

d.,').". 

—  0.0!) 

—  0.18 

|s(i-,'.(» 

-  1.  •.'.'. 

—  0.7S 

-  O.MT 

iN)(i.:» 

~    l.(M> 

—   ().(\!) 

-  0.73 

l^'KI..'! 

-  o.:J7 

—  0.10 

—  0.05 

Is  11.:. 

—  ().:n 

0."J(i 

—  O.'iH 

isl!)..-, 

—  o.:t7 

—  o.;U 

-  o.;n 

INJ4.S 

+  l.:.() 

+  1.4(1 

f    1.41 

1.S-MI.7 

-f  o.!)l 

-i-   0.!)0 

"i-  o.si 

is:{:).-.» 

—  0.'i7 

-  o.4;j 

—  ().4fi 

1n:$!).s 

-  0.17 

—  0.5(i 

—  0.48 

1S44.M 

-f  O.U 

—  o.:Jl 

—  0.18 

|S|!).«t 

-    O.'il 

—   0.70 

-  i)M 

|s.")l.!> 

-  «.I4 

—  i)M 

—  ().;}« 

IsiiO.l) 

—  M.l:{ 

—  o.*i:{ 

-  0.14 

|s(i,-,.(» 

+  *.:*« 

-f   o.io 

+   OM'i 

lHl((M 

—  »:*1 

-f    O.su 

-1    U.t4 

ir 


TIIK   ORBIT   OF   URANUS. 


173 


It  will  1)0  socn  th;it  the  effect  of  these  changes  of  weights  is,  that  the  older 
observations  are  a  little  better,  and  the  later  a  little  worse  represented.  I  concuivc 
tliat  our  choice  must  lie  between  the  first  and  third  solutions,  tiie  first  being  the 
more  probable  if  we  conceive  the  outstanding  residuals  to  b.;  due  to  errors  of 
observation  only,  and  tlie  third  if  we  suppose;  tlieni  e(pially  due  to  errors  of  com- 
putation. On  the  whole,  I  consider  the  mean  of  tin;  two  to  ho.  about  the  most 
probable,  and  this  will  giv(!  the  mass  of  Neptune  very  near  the  rouiul  nu.aber 

1 

i;>ioo 

which  will  be  adopter!  as    the  definitive    value.       T!ie  definitive   corrections  to 
Elements  III  (p.  i>'J)  will  then  be 


be'  {\X-W) 
h  (1N.>()) 


-  '.V'At 

-  li.4.> 

-  4.44 

-  4.1i 

-  O.'i.j 

-  o.i:n 


f'firi'trlioim  la  l/ii   Iinl'niutiua  and  A'w/c. 

These  corrections  have  been  derived  entirely  from  the  n\oderu  observatiems,  the 
ancient  oiu's  lieing  too  micertain  to  add  anytliing  to  tlie  weight  of  the  re-sidt.  The 
mode  in  which  the  correclif>n  to  tlie  latitu<le  el' the  \in)viNi(inal  epiiemeris  has  been 
conchuleil  from  fhi-  (ibservations  ims  been  sufliciciitly  explained  :  it  is  only  necessary 
to  add  tliat  the  immediate  vesidts  froiii  tlie  data  of  the  preceding  chapter  reepiirc 
two  corrections,  uanu'ly : 

(1)  A  correction  to  the  theoretieiil  latitude'  fi.r  the  change  in  the  adopted  mass 
of  Neptune.  The  value  of  tliis  correction,  as  derived  fioiu  the  data  of  ('liapt.r  \', 
is  witii  sulHcient  aiiproximatiou 

Jni  =  ()".'2.")7'cos(/. 

(•Jl  A  ceunclioii  to  the  observed  latitude  on  'ccouiit  of  the  difference  between 
the  oblitpiitv  (if  tlie  ediptie  adopted  in  tlii' various  epliemerides  compared,  au<l  that 
(if  UiDis.u's  Tnhhxilii  .Si,/,;/,  which  liaving  been  aelopled  in  the  tlii'or)  should  Ix; 
used  tliiiiugl!"ut. 

Applying  tin-  corree  tiim  {'J)  (l)lo  all  tlie  observed  bititudes,  we  hiive  the 
fidlowin<i  corrr,  tious  to  the  latitude  of  the  provisional  epheiiieris  derived  from  all 
tlie  oliscrvafi.-iis  of  e.uh  opiiositiou  since  HSl.  The  third  (olumn  gives  tlur 
number  of  ob-^i  rvatioiis  in  declination,  'i'liesc  nundters  may,  however,  in  somo 
cases  be  iuiu  luiat.'  Tlie  fourth  and  fifth  <  olinnns  givu  the  sine  ami  cosine  of  the 
ar-'umcnt  nt   latitude,  to  be  used  in  forming  the  eepiations  of  condition. 


174 


THE   O  11  B  I  T   OF   URANUS. 


in^i 


II  ii 


V^r. 

¥ 

No.  of  oUi. 

hill   H 

i'<i»  II 

Ytfiir. 

V 

Nu.  Ol'  ullH. 

•lu  u 

vol  i 

17M2.0 

—(1.4 

21 

+  0.31 

f  O.!!.'. 

1s:jo.7 

+  0.3 

4S 

— 0.M2 

—0.57 

n.s;i.o 

— ;i.'> 

13 

0.3S 

o.o:J 

1  S3 1.7 

(-1.0 

23 

0.«7 

0.50 

17M4.0 
i7s.').0 

—0.1 

13 
10 

0.5d 

0.  so 

O.Sj 

is.ii  7 
Is;t;t.7 

+0.0 
-to.n 

20 
f)4 

—0.90 
U.93 

—0.44 
0.37 

I7SM.0 

—  1.0 

5 

+0.71 

+  0  71 

1.S34.7 

+  0.3 

02 

0.95 

0.31 

17H!t.O 

H-(l.!l 

1    ,1   i> 

« 

0.77 

o.r,4 

1S3.-I.7 

+o.a 

71 

—9.97 

—0.24 

1  ilMI.O 
17',ll.0 

-1  •-'.(> 

-1  l.U 

4 

7 

0.s| 

O.St; 

0..'iS 

0..-.I 

ls.|(;.7 
ls:l7.S 

+  0.1 
0.0 

13.'i 
154 

0.98 
0.99 

0.17 
O.IO 

li'.fi.O 

-|1.S 

:i 

1  O.'.HI 

+  0  u 

Is.lS.S 

— 0.-2 

1S2 

—  1.00 

—0.03 

I7!l,».0 

ntu.o 

-f.'.'J 

3 

0.03 
0.  '.I.I 

o.:(7 
0  30 

is.m.s 

1S40.S 

—0  4 
—0.3 

142 
lOli 

1.00 
0.99 

+  0.03 
0.10 

Kitr.o 

17!t(;.0 

4<l.7 
-1  .•(.<•■ 

7 
4 

(0.07 
O.'.III 

4  <'.-•■-' 
oil 

1S41.S 
IsIl'.S 

—0  7 
—0.7 

lol 
li.'i 

— 0.9.S 
0.97 

+  0.17 
0.24 

17'.I7.(» 

4  .•i.i; 

3 

l.ou 

fO.II-l 

1SI3.S 

—0.0 

N.S 

0.95 

0.31 

is(m.2 

IMIM.-.' 

0.0 

+  I.-J 

i 

•-• 

|0.!IS 
0.07 

— 0  -.'l 

o.'ji; 

isll  0 
ls|.-,.0 

—  1.0 

— (I.e. 

H7 

—0.93 
0.90 

+  0.37 
0.44 

lso-.>.;i 

1  1-^ 

13 

0.  !l  i 

0.31 

ls4r..o 

—  11 

02 

0.S7 

0.50 

lH(ir>.;t 
iHO(i.:t 

-l-l.l 

—  1.4 

3 

r. 

-t(>.S3 
0.7S 

— 11.  "ii; 
o.i;3 

isn.o 

|sls  0 

—1.1 

— 0.0 

CO 

.'>(; 

— 0.H3 
0.70 

+  O.51; 
0.112 

lso7.;i 

-fl.C, 

d; 

O.Ti 

0.1,0 

Is  10.0 

—  1.0 

■M 

0.74 

o.(;7 

1X(IS.,| 

+  I.S 

n 

1  <t.r>7 

—0.74 

Is.Ml.'.l 

—0.7 

4l> 

— O.C.O 

40.72 
0.77 

l.s(i'.».;t 

+  l.;i 

•1 

O.l'.O 

(ISO 

1  s.-,  1 . 0 

—  1  .  •-• 

3."» 

O.IU 

IMIO.,! 

4  3.7 

K! 

o.:>3 

O.S.) 

1S.-.2.0 

—  1.3 

49 

0.59 

O.Hl 

Hll  .! 

-|:i.4 

11 

t  0  (.-, 

0.s!( 

Is.-,:!.o 

—  l.S 

4S 

—  0.53 

+  0.S5 

o.ss 

Mli'.4 

4;i.(; 

•; 

!>.:;>< 

(1.03 

1  s.i  1  0 

—  1.0 

47 

0.47 

lsl;i.4 

4-J.» 

t; 

O.JO 

0.01) 

Is, -,.-1.0 

—  1.3 

40 

0.41 

0.91 

|HU,4 

-1  •J.S 

i.t 

f  o.-.'i' 

— 0  OS 

Is.Mi.O 

—  1.1 

41 

—0.34 

4  (1.04 
0.97 

1M|.>.4 

■l:.'.!' 

ii; 

0  It 

0.00 

Is.iS.d 

—  1.7 

(;.') 

0.20 

isic.i 

-|l.:j 

H 

o.im; 

1.00 

IS.iO.O 

—  !.7 

fiii 

0.19 

0. 9» 

1HI7..'. 

-f.'.4 

13 

— 0  III 

— 1  no 

isr.d.o 

—  1.0 

M 

—0.12 

4  0.09 
1  110 

isis.r> 

+  I.J 

5J2 

0.110 

1.00 

lsi;i  (1 

—■'-.i 

41 

0.05 

1  Hi '.!.;, 

+  1.7 

7 

0.17 

0.00 

lsr.:!.o 

--1.0 

.'^.2 

40.02 

1.00 

lH;!l».r> 

+  1.0 

0 

— 0  I'l 

—0.07 

lsc,:(.(i 

—  1.7 

H3 

+  0.10 

+  0.99 

lx-.'l.'> 

+  l.:i 

10 

0  ::i 

O.O.'i 

Isr,  1.0 

■>-2 

39 

O.IS 

0.99 

l>s-2-2.:> 

+  1.S 

7 

0.3S 

O.OJ 

lsi;.'i.(i 

—  \.l 

37 

0.25 

0.97 

\x-2:t.:> 

+o.r, 

7 

—0. 1.^) 

— O.Sft 

l>-r,c,.i» 

—0.9 

72 

+  0.33 
0.40 

40.05 
0.92 

isoi.r. 

— 0.-2 

11 

o.r.i 

o.si; 

1  si;  7. 11 

— 0.(> 

«3 

lH->:,.:, 

+  1.5 

5 

0..-.7 

O.si 

ll'.CS.l 

—0.0 

32 

0.47 

O.SS 

IH  •.'(;.  5 

+  1.-J 

7 

— o.r.:{ 

—O.TS 

lsr,!l  1 

— o.s 

(i.'i 

+0.54 

+  0.S4 

H:J7.(; 

-1  1>.0 

■I 

0  I'lS 

(1.73 

IsTO.l 

—0.4 

32 

o.i;o 

o.so 

Is2s.(i 

—  l.h 

7 

0.73 

o.r.s 

IS71.1 

—0.7 

21 

4  0.(;(i 

0.75 

ls2".t.7 

i 

+(!.!» 

9 

— 0.7S 

— O.f.3 

Is7:J  1 

—0.4 

47 

+  0.72 

+0.(19 

It  will  be  rcmomlicrcd  tliut  the  oliscr^'orl  (Iccliiiiitions  have,  ns  for  11s  imssililc, 
liccn  reduced  to  .Viiwers'  .standard.  \N'e  li.ivc  no  |Misitive  proof  that  this  »<taiidard 
is  I'orrcct.  If  it  Ik;  affected  hy  a  coiistaiit  error,  tlie  residt  will  he  that  tlii'  orhit 
(if  the  planet  011  tlic  celestial  sphere,  as  didiiced  from  oliservatioii,  instead  of  hein^ 
a  i^nNit  circle,  as  wa  know  the  real  orhit  to  he,  will  he  11  small  one,  and  the  com- 
jiarison  of  a  uniform  series  of  observations  evteiidin^'  tliro!i;:h  an  entire  revolntiim 
of  the  planet,  after  niakin;;  tlie  be>t  correction  to  tlie  position  of  the  orhit,  will 
leave  a  constant  residual.  Now,  we  can  best  ihtrriiiine  lliis  residual  by  includinjj 
it  us  uu  unknown  ipiantity  in  oiir  eipiations. 


THE  ORBIT  OF  URANUS. 


1 


i-> 


Aj^uin,  the  error  of  the  stnndnril  is  not  nrccssnrily  coiistniit,  Imt  miiy  contiiin  a 
term  iiriiporliuiial  tu  tliu  tiiiii',  arising  from  erroneous  proper  motions  of  tlie 
stundaril  stars.  'I'lierefore,  imstead  of  siipposing  tlie  residual  eoiiMtaiit,  we  shall 
suppose  it  of  the  form  a  -j-  hi.  Kaeli  observed  correction  to  the  theoretical  latitude 
will  then  give  the  cqiuition, 

bin  «.^<^  —  cos  uiphO  -\-  a  -(-  h  7'=  h^i. 

To  facilitate'  tlie  solution  of  these  e(|natioiis  they  have  been  di\ided  into  groups, 
each  group  usually  comprehending  three  oppositions,  and  comliined  into  a  single 
etpiatiou  multiplied  hy  such  a  factor  as  would  make  its  proltalde  error  half  a  second. 
'I'Ik!  factor  hy  which  the  correction  of  the  latitude  is  multiplied  in  the  equation  is 
the  same  with  tin;  coeflieieiit  of  ii.  The  year  1H4().0  is  taken  as  tin-  epoch  for  A. 
'I'hus  we  have  the  followin*': 


KglATIciNS  or   I.ATITI  UK. 

t)iitf:<  of  o|i|iniiHlnnii. 

No.  of  op|>. 

KlIIMlllMI. 

ITMS.O-S.l.d 

4 

0   i^f 

— o.'.ifi'y          1  1  .On 

— n.fifc 

:    —\.r, 

I'SS.O-'.II.O 

4 

OS 

— o.i;          ^  1.0 

— O..W( 

..    4  1.1 

I7'.>2.0  Itt.O 

:i 

o.."> 

—0.2             -|0.5 

—0.2 

^      -lO.!! 

17!l.">.0   !I7.I 

;i 

a.:! 

0.0           4  »."> 

—0.2 

.      p».7 

isno  2  o-.>  .1 

;i 

(1.0 

■fO.IJ              -fl.O 

—0.4 

0.  4  1.0 

lso.''i,;t  07. :l 

:! 

o.s 

-1 0.(1       ^  1.0 

—0:1 

=  4  "7 

|HOM.;j  10. :i 

;i 

0.(1 

-I  0..S                 1   1.0 

— o.;i 

-  4  2.(1 

isii.:t-i.:  J 

;i 

0.4 

-1  0.9               1  1.0 

—0,1 

-  4:1.1 

isn.»-n;.  J 

.•t 

0.2 

•  tl..'')            -11.;^ 

—0.4 

:           f.TO 

isi7.(-i!i.ri 

:( 

—0.1 

•  fl..'-)            +1..'-. 

— o.:i 

1  4.S 

1S20..'.  •Jl'..''> 

;: 

— o..'> 

4  1.4            4  !.•'■. 

— o.:i 

-     12.0 

is-.';i. .'■>-•-'.■)..". 

:! 

-~0..'> 

■1  O.ft                 1  1.0 

—0.2 

>a     1  0.(1 

ls-_'r,,.'i  I's.c, 

>*i 

-0.7 

-1  0.7             i  1.0 

—  0  1 

-     I  0.(1 

1  s  I'll,  r,;;  1.7 

•■t 

-1.2 

.|0.S               -1  l.Ti 

—0.1 

-  4  1.1 

ls:!J.7  :;i  7 

:i 

-^2.S 

-|l.l          -|:i.o 

—0.2 

-^:-\\.^ 

Ih:-,;-,  7-::7.s 

;{ 

— 2.;» 

1  O..'.               -)  .'1.0 

—0.1 

-.  -,  o.:i 

|s:js.,s-io.s 

;i 

-;!  () 

—0. 1           +:i.o 

0.0 

m,  0  !) 

isil  M-|;;.s 

••1            — -J.'.i 

-0.7           4:1.0 

4(U 

—  — 2:1 

isu.s.ic.s 

.! 

—•-'.7 

-l.:i          -1  ;i.o 

4i».2 

=  -2.7 

1S47  S-t!M> 

•> 

— 2. « 

—  1.!)           -1:1.0 

4  o.;i 

:    — ;i.o 

\<A).'.)   :rl.'.) 

•» 

—  l.'.l 

— 2.:i          -f:io 

40.4 

=.  — :j.2 

ls.-,:t.i(  aU.W        '         ;! 

—  1.1 

—2.1!             1  :i.o 

4  0.4 

=  —4  1 

|m,m;.!(  ."I'.i  0       j        :i            — o.s 

--2.0           i  :i.o 

4  0..-. 

=  —4.5 

isr.o  0  i;-.'.o               :;            —n.-l 

— :;.o           1  ;t.o 

-•  (I  11 

;    — r).s 

isc.i.o  i;.'..o              :!             1  o..'! 

— :i.o           4  :i.o 

•io.T 

=  — .^.(! 

iscc.o  r,s  0               .1 

1  I.-' 

— 2.S           4:1.0 

4  o.s 

—  — 2  4 

isc,'.).!   70  0               •_' 

^  1.1 

— 1.(!           42.0 

-1  0.(1 

=r^— 1.2 

1S7I.I    7i'  1         1         -• 

-i  i.« 

-   1.4             I  •-'.<> 

4  o.(i 

^:— l.l 

Treating  these  eiiiiatioiis  liy  tlir  iiictlnxl  of  least   s([iiiiiis,  we   find   the  normal 
e({uations 

If  It  n  'I  X 

():5.C.0,^./)  +    {\.\ruf„\0—    rri.li),,,—    l.OSAr-.  4.    t>s.;{:l 
(;.  l.>.^<^ -I    (i!»..'i!>r/»V)  —    .V.MiO.e  —  1  I. :.■)/<         I    lll.d'i 


li 


'it 


1.O8A0  -    1  l.'v';"»</«N«  -j-       H.\K„t  ■{■     i.l\)h 


•i'.i.iW) 


11 


no 


Til  i:  o  It  II I  r  o  I''  II  It  \  N  II  s. 


ir 


Till'  hiillllloll  lit    llii'sr  ri|ii.illii|ii  ^i\i's 

•V  I    0    •.'N    I    (I  .1.1   ,1  I    II    ,i| 

yiAl'  I     I  .^I't     I    0   liMliii     I    ll.'.'ll.iA  I     I     i.i 

II       I  n  .:i.i 

/.  II    V'S 

Tin  >.i'  tiliii'N  III'  •(  .mil  /'  lliilli  nil  III. it  lit  ihr  .  |iiirli  |M|II  \invrls'  i'i|l|iiliil  llll 
llrrllllMtliillN  nil'  tiMi  en  Ml,  III  lir<  liHltll  |iiil<il  ili'iliiiii  r-.  iilr  Inn  slililll  li\  U".:i,'),  jinil 
I  hill  tlilHi'irni  II  til  III  III  isli  111)^  III  till  1. 1 1 1'  III  O  .'.'M  |ii  I  ri'iiliii>.  ||  tlir  nliln  liiniNiiir<4 
III  ilri  lllliilliill  liiiil  lirrli  i  ii||i|MIMl)lr  III  |ilri  IMiHl  U  llli  IIiiim'  niiiilr  ill  till'  |Mrsi'lll. 
lillir,  Mini  ll'  till-  |iiisMli|r  iii'iiKilii  riliil  in  llir  irilmril  ll)^llt  iiii  rlisiiiiis  linij  lirni 
Clllrlllll)  rlllllllliilril,  I  sliiiiilit  ir;;.ilil  llii  .  ililrl  liiMliltliill  lis  rlilltliil  tii  riMINJlli'l'illlli' 
Mi'l;;lll  III    \lr«\    iit'    lltr   <;ir,ll    lllli  rl  t.illll  \    ill     llir  ilri  ll|l;|l  Inlis   |i|r\|iM|-i    to    IN'.'II,   it 

run  III'  n'l^.inlril  ns  liiilr  mmi'  lli.in  u  iiiii.';li  iillrin|il  iil   ii  ilrlriniiniilinii.      I'm  lliii 
Irii-tiin  till'  lii-^l    h\ii   iimniiil   ri|n.itimis   Iium'    lin  ii    shImiI,  Ii.uiii;;  h  imil  !•  milrlri 
ininiitr,  so  IIS  III  hliinv  tin    \iiIm  s  nl   ,\i^,  miil  i/iAll   in    Iriins  nl    tin  sr  i|niiiitilirM       ||. 

Mill  III-  Nirtl  lllilt  llilil  Ki'  ni';;li'i  till  .1  iinil  /<  rnliii'U,  till'  tilllli'  III'  Ai/i  umilil  IllMr 
lirrll  stiliillrl  li\  0  '.'Ii,  Mini  llnit  nl  i/i,\|/  Mihillri  li\  IT  |S  tllMli  IIiiim-  iiiIiiiIK  inn 
rlllili'il  \'.  llir  nliM'M  iitimis  »  itli  tlir  N\  ,isliin;;lnn  Tl  iin'.ll  ('llilr,  iinil  tllnsr  N«|||| 
lln'  rnlKiMtii  \  rilii  ill  (  'mil',  Imlli  liiilii.itr  im  linir.i'.i'  nl'  \in\il't'  |iii|.il  ill'.liinri  s, 
I  --ll  ill  hiKi'  Im  llir  ill  lililllM'  I  I'l  III  llnii'i  In  I  lir  lIli  llll.lt  Inll  imil  liiiili'  lllir.i'  u  lili  li 
inlinn    limn  tin'  iilin\r   \,ili|i I'  ii   iilnl  /',  m. 


W\ 


.Nl/. 
i/..N() 


I    II     ll 
I     I     ..-., 


Tlic  rnllnwilli;  l.ililc  -.li.nv^  tin'  ii-.ii|ii,iK  nl'  llii'  i'i|ii;ilinns,  miiiI  tin'  iiii'mii  mil- 
Ntiinilllif^  iniii  rlimis  to  tin-  l.ililinli  ,  \  I  |  win  n  tin'  i  i.ni  liiili  i|  \  .\\\\i  ;  nl  .N</>  miiiI  ./..Vl 
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||>i'   |i'|iri'si  liliillnll  III     llir  l.li.ilMltli.llS. 

I  liiiM'  nut  iilli  iii|ili  il  II  ti;'iiinii'.  in\i  slii'iiliiiii  iif  lln  |.ri.l.Hlilr  iiini  iif  miy  of 
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iiii'tlinil  id  II  Mst  si|nmrs  Hi.iilil,  III  a  iiisr  liKr  llir  |iii  .i  III,  lit  iniiri  ly  nnliiiNlwnrlliy. 

II  is,  IliiWiM  I      Ml\   ill  Mialilr   Ihal   \M'  '.lliilllil    \»-  alilr   III  riillll   s •   jl|il);nirlll   III'  llli- 

niiii  il  linU  111  tlir  iini'.^.  nl'  Nr|.lmir.  I'mni  llir  la. I  sy.li  in  nl  i  i|iniliiiiis  ul  ininli- 
tliill  llir  >alilr  nl'  »' nillli'H  mil,  willl  llir  Wril-lll  :i  lit,  ni  liiailv  lllil  liNsij^'lli  il  In  llir 
nil  III  naill  nl'  i  aill  IiM'  \ial-.  i.I'  iimililli  nl.srivalinlis.  Ki7,'aiillli(,'  lliisr  riMllls  iih 
null  |ii  lull  III,  llii  II  nil  all  i mil  Mnlllil  lir  al.iilll  0"..'.,  sn  llial.  llir  |i|iilialilr  rum  nl  ^,' 
Wiilllil  lir  0..,  anil  llial  nl  ii  »»nnlil  lir  00...  nr  alinlll  ,  ,',  „  llir  ilililr  Iiiiisn  nl" 
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III  1  II    ininll    s Ill  T.  I    wlirll,   ill   llir   la. I   rijiial  lull  .,   \Vr    allow   lor    llir    s\.lrlliall(; 

«  liailM'Irr  nl"  llir  rrsiillials,  il  mil  lit  Im^n.  IT  \Vr  sli|i|.iisr  llir  llirnl)  'n  lir  iiill'iil, 
I   n.lirrivi'  \vr  iniiy  I'ailh  i  slimali    llir  |ili.lilliilr  i  itin   III    llir  inil-iM  III'   Nr|.liiiii    In  lir 

>       III'  ils  I  lilllr  miliiinil.  anil   ils  |.nssili|r  rlliil    lun  nr  llin  r  linirs  (.'iratrr.      H    Mil  ir 

JM  any  ninr  nr  iniiirrlri  Imii  in  lln    llimi),  lln   i  iinr  may  In    niinli  lai;.;i  i. 


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33  WEST  MAIN  STRIi  T 

WEBSTER,  N.v      •tSSO 

(716)  873-4503 


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178. 


THE   ORBIT   OF   UUANUS. 


CHAPTER    VIII. 

COMPLETION  AND  ARRANGEMKNT  OF  THE  THEORY  TO  FIT  IT  FOR 

PERMANENT  USE. 

In  the  preceding  discussions  the  terms  of  the  second  order  due  to  the  action  of 
Neptune  have  been  neglected,  the  elements  of  Uranus  and  Neptune  being  so  chosen 
that  these  terms  can  scarcely  become  sensible  within  a  century  of  the  epoch.  But 
this  very  choice  will  make  them  larger  in  the  course  of  centuries  tlian  if  mean 
elements  had  been  chosen.  They  will  be  most  sensible  in  the  case  of  the  great 
inequality  of  4300  years  between  Uranus  and  Neptune,  an  inequality  which  will 
make  centuries  of  observation  necessary  to  an  accurate  determination  of  the  mean 
elements  of  the  two  orbits.  The  uncertainty  arising  from  the  great  inequality  is 
probably  of  the  same  order  of  magnitude  with  the  omitted  terms  of  the  second 
order,  and,  such  being  the  case,  the  theory  would  really  be  made  but  little  more 
accurate  by  the  addition  of  those  terms.  1  conceive,  however,  that  the  theory  will 
be  made  much  more  satisfactory  by  the  computation  of  at  least  the  largest  of  the 
terms  in  question,  if  only  to  arrive  at  a  certain  determination  of  their  order  of 
magnitude,  and  of  their  effect  on  the  planet  during  the  period  in  which  it  has  been 
observed. 

The  term  in  que  lion,  being  of  very  long  period,  may  be  most  advantageously 
treated  by  the  method  of  variation  of  elements,  more  especially  as  it  has  in  the 
theory  been  already  treated  as  sucii  a  perturbation.  The  largest  of  the  pertur- 
bations in  question  are  those  of  the  mean  longitude  which  arc  multiplied  by  the 
square  of  the  integrating  factor  r,  which  is  nearly  51,  but  which  also  contain  the 
eccentricities  as  factors,  and  those  of  the  eccentricity  and  perihelion  which  are 
independent  of  the  eccentricities,  but  are  multiplied  by  only  the  first  power  of  j; 
Tliese  terms  will  probably  comprise  nearly  or  quite  nine-tenths  of  those  arising 
from  the  term  of  long  period. 

Let  us  begin  with  the  perturbations  of  mean  longitude.  These  are  given  by  the 
integration  of  the  equation 

-^^  =  —  ^m'ati'  \  r/.-,  sin  (2Z'— ?— tt)  +  c%  sin  QlT—l—n')  \ 

l\  and  /i-j  being  functions  of  the  ratio  of  the  mean  distances,  or  a.  If  we  integrate 
this  equation,  supposing  all  the  quantities  in  the  second  member  except  /'  and  I  to 
be  constant,  and  these  two  to  be  of  the  form  nt  -(-  f,  n  and  e  being  constants,  wo 
shall  reproduce  the  principal  term  of  long  period  already  found.  But  in  the 
second  approximation  we  must  suppose  all  the  elements  variable.  It  is  not,  how- 
ever, necessary  to  take  into  account  the  variations  of  «,  n,  and  /r,  bccau:,c  these  are 


THE  ORBIT  OF   URANUS. 


179 


fi  within  the  brackets  will  thus  become 


of  a  lower  order  of  magnitude.     The  perturbations  to  be  added  will  be  those  of 
I,  I,  e,  e,  7t,  and  n. 

The  point  from  which  the  longitudes  are  counted  being  arbitrary,  we  shall  take 
the  position  of  the  perihelion  of  Uranus  for  1850.0  as  the  origin,  and  put,  as  before, 
g  for  the  mean  longitude  '<f  Uranus  counted  from  tliis  point,  and  let  I'  represent 
the  mean  longitude  of  Neptune  counted  from  the  same  point.  The  terms  of 
fl 

de 

eky  sin  i2l'—ff—bn)  +  eV.-,  sin  {2l'-g—{n'—7t)) 
or,  if  we  put 

<;'  sin  (ti' —  7t)  =  h' 
e'  cos  (ti' —  7i)  =  // 

and  notice,  that  to  terms  of  the  first  order  we  have,  sin  ^,n  =  hn.,  cos  Sn  =  1,  we 
shall  have 

^~  =  —  3w*'a»r  \{cl;  +  Jc/,;)  sin  X—  {ckM  +  h'k,)  cos  iV^| 

differentiating  the  quantities,  of  which  the  perturbations  are  to  be  considered  with 
respect  to  the  sign  b,  we  find  for  the  terms  of  the  second  order. 

S   J,   =  —  3m'a»"  I  (/.\Se  +  /.-j-V.' +  7t7.-jSiV^)  sin  iV 


d£ 


-|-((eZ;i  +  Jc'h)  hN—eki^Ti  —  k.,^7i')cos  Nl 


We  have  now  to  substitute  in  this  expression  the  numerical  values  of  the  quanti- 
ties within  parentheses.  Those  of  the  perturbations  of  Uranus  have  already  been 
given  in  Chapter  III,  but  it  is  necessary  to  diminish  them  by  the  factor  0.145*  for 
the  altered  mass  of  Neptune.  Those  of  Neptune  are  taken  from  my  investigation 
of  the  orbit  of  that  planet  (p.  38).  The  mass  of  Uranus  there  adopted  is  ^j^-^-^^ 
while  the  investigation  of  Dr.  Von  Asten,t  from  the  observations  of  Struve  and 
others,  shows  it  to  be  tttj Jjt^.  The  perturbations  are  therefore  diminished  by  ^\. 
In  accordance  with  the  system  adopted  throughout  both  investigations,  constants 
are  added  to  all  the  perturbations  to  make  them  vanish  at  the  epoch  1850.0.     A 

terra  is  also  added  to  make  '    also  vanish  at  the  epoch ;  this  corresponds  to  the 

dt 

constant  which  ought  to  be  added  to  hi.     The  numerical  values  thus  obtained,  are : 


*  This  factor  was  adopted  before  the  niasa  of  Neptune  to  be  employed  had  been  finally  decided 
upon.     Hence  the  difference  between  it  and  that  in  the  preceding  chapter, 
f  Mdmoires  de  rAcaddmie  de  St.  Tdtcrsbourg,  tome  xviii,  vii  sdrie. 


J| 


180 


Till   ORBIT    OP   UUAJiUS. 


8N=  + 

c57t   =    — 
^C     =   - 

Ih'  =  4- 
hfc  =  4- 

/.-,  =  —  1.234 

k^  =  -j-  0.452 
h'  =  4-  0.00695 
Jc  =  —  0.0048G 


72G0"  sin  N—  6658"  +  4".26< 
414  smiV+    380 
414  cosN—    165 
120  sinJV—    110 
120  COS.V+      48 


fin 


{ft 

Substituting  these  values  in  the  expression  for  5-,.2    and  integrating  twice,  we 

find,  putting  b  for  the  coefficient  of  the  time  in  iV,  of  which  the  value,  taking  the 
century  as  the  unit,  is  -f  0.1472,  and  putting  Tfor  the  time  in  centuries. 


il=  „  ni'av^ 


|(411"  + 


1  no"  8  '»"  'X 

-  J  •    +  2".6  T)  sin  iV  +  (1837"  +  'Y  -51."4  ^  cos  N 


—  109".3  sin  2iV  —  5".5  cos  %N  \ 
—  16".5»j'a»t¥4-c7'+c', 
c  and  c'  being  the  arbitrary  constants  of  integration,  which  are  to  be  chosen  so  that 
both  U  and  its  first  differential  coefficient  shall  vanish  at  the  epoch.     Reducing  to 
numbers,  we  find 

5?  =  (140".70  +  0'.327')  sin  iV 
+  (232.60— 6. 377')  cos  iV 

—  13  .60  sin  2JV 

—  0  .70  cos  2.V 

—  0.037^^ 
+      34  .27  T 

—  46.76, 
the  last  two  terms  being  arbitrary. 

When  we  carry  the  perturbations  of  the  eccentricity  and  perihelion  to  quantities 
of  the  second  order,  we  are  troubled  by  the  introduction  of  large  terms  depending 
on  the  square  of  the  disturbing  force,  which  disappear  from  the  rigorous  expres- 
sions for  the  co-ordinates.  These  may  be  avoided  by  substituting  for  the  eccen- 
tricity and  perihelion  the  quantities  h  and  Ic  determined  by  the  condition 

/t  =  e  sin  rt 
k  =  c  cos  n 
If,  as  before,  we  count  the  longitudes  from  the  perihelion  of  Uranus  at  the  epoch 
1850,  we  should  substitute  ^n  for  n  in  these  expressions.     The  values  of  h  and  /.; 
will  then  be  given  by  the  integration  of  the  equations 

*''*  '         7  XT  • 

-J   =      matiiCi  cos  N 


dk 
lit 


=  —  ni'ank,  sin  N. 


THE   OllBIT   OF   UUANUS. 


181 


Differentiating  with  respect  to  ^,  we  find  for  the  terms  of  the  second  order 


5  ,    =  —  ni'anhi  sin  Nh  N 
lit 


.(Ik 


—  vi'anJii  cos  Nf)  N. 


Substituting  for  hNits  numerical  value  just  given  and  integrating,  wc  find 
ih  =  m'uvJci  \  —2895"  sin  .V—  (G658"  —  -l^'iG/)  cos  iV+  1815"  sin  2.V| 

—  3630"/«'a/i'i«<      -j-  constant ; 
tk  =  m'avh\  1-2895" cosiV+  (6658"— 4".2G0  sin  iV^+  1815" cos  2.V| 

-|-  constant; 

the  constants  being  so  chosen  that  <Vt  and  hic  shall  vanish  at  the  epoch. 
Reducing  the  values  of  hh  and  hk  to  numbers,  they  become 
tJi  =  5".82  sin  iV+  (13".40  —  0".867')  cos  N—  3".65  sin  2iV+  r'-OST—    2".61, 
hk  =  5  .82  cos  N—  (13  .40  —  0  .86  T)  sin  N—  3  .65  cos  2.V  +  12  .12, 

the  last  two  terms  being  arbitrary  constants. 

Computhig  the  values  of  these  terms  of  fil,  <Jt,  and  hk,  for  intervals  of  50  years, 
from  1600  to  2000,  we  find  them  to  be  as  follows: 


Year 

n 

,7t 

hk 

1600 

—V.U 

+0M0 

— 0'.02 

1650 

—0.71 

+0.05 

0.02 

ITOO 

—0.31 

+0.02 

—0  .01 

1750 

—0.10 

0.00 

—0  .01 

1800 

—0  .01 

0  .00 

0  .00 

1850 

0.00 

0.00 

0.00 

1900 

0.00 

0  .00 

0  .00 

1950 

-j-0  .04 

0  .00 

—0.01 

2000 

+0.18 

—0  .01 

—0.02 

We  see  that  although  the  ultimate  effect  of  these  terms  is  very  considerable, 
their  effect,  during  the  period  that  Uranus  has  been  observed,  is  insignificant. 

Conchided  Elements  and  Perturhntions  of  Uranus. 
The  corrections  found  in  the  last  chapter  being  applied  to  the  final  provisional 
elements   (p.   99)   give  the  following  elements  for  1850,  affected  by  the  great 
inequality  produced  by  Neptune: 

Elements  IV  of  Uranus. 

Epoch,  1850,  Jan.  0,  Greenwich  mean  noon. 
168°    15'      6".7 
28     25     17.05 


0, 

■i|^ 


73     14       8.0 
0     46     20  .54 
.0469236 


182  THE0RBIT0PURANU8. 

c  (in  sec),  9678".69 

n,  15425.752 

log  a,  1.2829072 

log(/„  1.2831044 

Log  «,  includes,  as  before,  tlie  constant  term  in  the  perturbations  of  the  logarithm 
of  the  radius  vector  which,  with  tlic  corrected  mass  of  Neptune,  is  +.0001972, 

To  find  the  correspoi.  ding  elements  at  any  other  epoch,  the  following  secular 
and  long-period  perturbations  are  to  be  applied.  Those  produced  by  Neptune  are 
derived  from  the  expressions  in  Cliapter  III  by  correcting  them  for  the  new  mass 
of  Neptune,  and  for  the  change  in  the  value  of  tlie  small  divisor  2n'—u  j/roduced 
by  the  correction  of  the  elements  of  Urauns.  The  logaritiims  of  the  factors  for 
correction  arc. 

Correction  of  mass  of  Neptune  9.93598 

Correction  of  divisor  0.00051 

Log.  factor  for  H  9.9349G 

Log.  factor  for  le,  Sn,  5»  9.93547 

Including  the  perturbations  of  the  second  order  just  found,  we  have,  by  putting 

N=2r-!,, 

=  1 13"^  30'  4G".0  -f-  8°  2G'  51".9  T, 

tl  =  (-2850".41  +  0".32r)  sin  N-\-  (387".G7  -  G".377')  cos  iV 
+  112  .72  sin  2.V  —      47.28  cos  2N 

—  7.72sin3iV  +        4.33  cos  3.V 
+       0 .55  sin  4iV                 —        0.4G  cos  4Ar 

—  0  .037'==  — 83".787'+  2811  ".41. 

M  =  —  412".18  sin    iV+  (14".03  —  0".8G7')  cos  N 
+    29.20sin2.V—     G  .09  cos  2iV 

—  3.11.sin3.V+  1.19  cos  3.V 
+  0.28  sin  4.V—  0.13  cos  4.V 
+    14.7G7'  +398.33 

a-  =  —  41 1".53  cos    N—  (13".G5  —  0".8G  7')  sin  N 
+    29.33cos2iV+     G.17sin2iV 

—  3.12cos3.V—  1.21  sin  3iV 
+      0.29  cos  4.V+     0.13  sin  4iV 

—  5.4537'        —124.72 
3n  (in  units  of  tlie  7th  place  of  decimals). 

=  1963  cos    iV^+103sin    iV 

—    171cos2iV—    G7sin2iV 

+      15  cos  SAT  +      Gsin3iV 

+  511.0. 

The  perturbations  of  e&n  and  le  are  here  replaced  by  those  of  h  and  k,  defined 
by  the  equations 

7i  =csin  {7t'—n„) 
k  =  e  cos  (ti  —  rr,) 


.-1-^ 


THE   ORBIT   OF   URANUS. 


183 


7to  representing  the  perihelion  of  1850  =  168°  15'  6".7.     We  then  have  Jor  tlie 
eccentricity  and  longitude  of  perihelion  at  any  epoch 

c  sin  (n;  —  7t„)  =  f,h 

e  cos  (7t  —  TTo)  =  Co  -\-  fil: 
In  the  above  terms  multiplied  by  the  time  we  have  included  the  sectdar  varia- 
tions produced  by  Jupiter  and  Saturn.     If  the  perturbations  of  the  elements  due 
to  cacli  particular  planet  are  required,  we  have 

Action  of  Jupiter, 

h7i  =  +  5".  737';  hk  =  —  0".G087'. 

Action  of  Saturn, 

,7i  =  +  5"MyT;  th-  =  —  4.589 f. 

Subtracting  these  from  the  above  exjjrcssions  all  the  remaining  terms  will  be  duo 
to  tlic  action  of  Neptune.  The  values  of  U  and  bv  are  due  entirely  to  the  action 
of  Neptune. 

For  the  sake  of  rigor,  we  may  suppose  the  perturbations  produced  by  each 
planet  to  be  multiplied  by  a  factor  representing  the  number  by  which  tlie  adopted 
mass  of  the  jdanet  must  be  multiplied  to  obtain  the  true  mass. 

It  will  add  to  the  homogeneousness  of  the  tlieory  to  express  the  i)orturbations 
of  long  period,  wliich  aw  multiplied  by  the  product  of  the  masses  of  Jupiter  and 
Saturn,  as  perturbations  of  the  elements.     These  terms,  as  found  on  page  88,  are 

{v.c.O)  =  —   0".55  sin  N^  —    0".()3  cos  iV,, 

+  40  .(iu  sin  iV,  —  10  .50  cos  iVj 
(r.«.l)  =  -|-    2  .(54  sin  iV,,  +    4.()4cosjVo 

-f  7  .35  sin  iVj  +  4. 41  cos  iV^ 
(r.f.l)  =  —    4  .'.>3  sin  iV„  —    3  .87  cos  .V^ 

+    8.06siniVj—    8. 38  cos  .V, 

These  terms,  together  with  the  arbitrary  corrections  of  the  elements  whi'ih  have 
been  applied  to  make  them  very  small  at  the  epoch,  may  be  replaced  by  tin  follow- 
ing corrections  to  the  elements: 

,y  =  —   0".55  sin  iV,,  —    0".03  cos  N, 
+  40  .05  sin  iV,  —  10  .50  cos  N., 
4--27".-27  —  ll".7-.>7'. 
,Vt  =+    2  .09  sin  .V,  +    1  .94  cos  N^ 
'  —    -i.lSsiniV,-}-    3.71  cos  iV^7 

+  1".'28. 
a- =  4-  1  .3'2  sin  iV„ -1-    2.32cosiVr„ 
4-  3  .08  sin  iV,  +    2  .21  cos  JV, 
fiv  =      27  sin  iV,  +  i04  cos  iVj  +  76  (in  luiits  of  the  7th  decimal). 

The  amount  of  the  perturbations  of  the  elements  for  every  half  century,  from 
the  year  1000  to  2200,  is  given  in  the  following  table.  Cohinm(l)  gives  the  per- 
turbations  by  Neptune,   Saturn,    and   Jupiter,  computed    from    the    expressions 


m 

1' 


m- 


m 


184 


THE   OR  BIT  OF   URANUS. 


on  pago  182;  column  (2)  those  just  given  depending  on  tlic  product  of  the  masses 
of  Jupiter  and  Saturn. 


Year. 

hi 

Ih 

hk 

hv 

hx 

h 

^'\, 

C?) 

(•>, 

(2) 

^'l 

(2) 

(!) 

(2) 

II 

1000 

+2050.31 

+100.09 

+ 

42.00 

+1.94 

—389.51 

+G.08 

+  1955 

—140 

—2.00 

+  1.20 

1050 

1841.17 

149,08 

27.79 

3.00 

375.05 

0.08 

1882 

150 

2.10 

1.37 

1100 

l(i38.70 

130.39 

14.03 

4.99 

300.31 

0.82 

1802 

169 

2.12 

1.64 

1150 

1444.87 

122.87 

+ 

1.45 

0.01 

343.49 

6.47 

1717 

178 

2.08 

1.60 

1200 

1200.24 

108.85 

9.90 

0.00 

325.27 

5.04 

1026 

183 

2.02 

1.77 

1250 

+  1085.7(i 

+  94.07 

— 

19.84 

+  0.75 

-305.09 

+4.42 

+  1529 

+  183 

—1.94 

+  1.80 

1300 

921. 7(i 

80.07 

28.34 

0.49 

284  90 

2.80 

1420 

180 

1.84 

1.91 

1350 

7(!9.32 

07.12 

35.27 

5.90 

252.90 

+  1.10 

1318 

173 

1.70 

1.92 

1400 

fi29.00 

54.35 

40.50 

5.08 

239.78 

—0.57 

1205 

101 

1.64 

1.91 

1450 

501.39 

42.02 

44.11 

4.16 

215.08 

2.17 

1080 

147 

1.37 

1.85 

1500 

+  387.18 

^-  32.11 

— 

45.83 

+3.24 

—190.00 

—3.52 

+  903 

+  129 

—1.19 

+1.74 

1550 

28G.05 

22.78 

45.03 

2.42 

104.84 

4.52 

835 

110 

0.99 

1.60 

1(100 

200.  fi5 

15.38 

43.40 

1.80 

138.31 

6.12 

704 

90 

0.80 

1.42 

1G50 

129.43 

9.34 

39.20 

1.41 

111.22 

5.30 

608 

08 

0.01 

1.20 

1700 

73.40 

4.80 

32.79 

1.23 

83.70 

5.09 

430 

48 

0.44 

0.95 

1750 

+     33.00 

+     1.90  — 

24.10 

+  1.23 

—  55.90 

4.50 

+  288 

+  29 

—0.27 

+0.60 

1800 

8.51 

0.33  _ 

13.20 

1.30 

—  27.93 

3.81 

+  146 

+   11 

—0.12 

+0.34 

1850 

0.00 

0.00 

0.00 

1.52 

0.00 

2.00 

0 

—    5 

0.00 

0.00 

1000 

7.04 

0.71   + 

15.05 

1.03 

+  27.74 

2.10 

-  145 

18 

+0.10 

-0.30 

1950 

31.47 

2.23 

33.70 

1.59 

55.09 

1.34 

290 

26 

0. 10 

0.73 

2000 

+     71.43 

+     4.25   + 

54.20 

+  1.34 

+  81.81 

0.73 

—  433 

—  32 

4  0.20 

—1.10 

2050 

127.33 

0.52 

77.12 

0  85 

107.77 

0.34 

574 

30 

0.20 

1.47 

2100 

198.80 

8.75 

102.40 

+0.13 

132.07 

0.14 

711 

27 

0.17 

1.83 

2150 

285.72 

10.59 

1.30.15 

—0.77 

150.33 

0.11 

843 

20 

0.11 

2.17 

2200 

387.11 

+  11.77 

+ 

100.15 

—1.70 

+  178.48 

—0.19 

908 

—     9 

+0.02 

—2.48 

Mean  Elements  af  Uranns. 

If,  instead  of  the  elements  of  Uranus  affected  by  the  great  inequalitj',  we  wish 
the  absolute  mean  elements,  these  are  to  be  obtained  by  adding  to  the  elements 
already  given  the  constants  applied  to  the  perturbations  hi,  hh,  hk,  and  hn  to  make 
the  perturbations  vanish  at  tiie  epoch  1850.0,  and  also  the  corrections  (p.  113) 
which  we  have  subtracted  from  the  elements  and  added  to  the  perturbations  to 
reduce  the  latter  to  a  small  quantity  during  the  period  for  which  the  tables  are 
likely  to  be  used.     We  thus  find  the  following  mean  elements : 

Elements  V  of  Uranus.     Epoch,  1850,  Jan.  0,  Greenwich  mean  noon. 


l/ongitude  of  the  jierihelion 

Mean  longitude  at  epoch, 

Longitude  of  the  node, 

Inclination  of  the  orbit, 

Eccentricity, 

Eccentricity  in  seconds, 

Mean  motion, 

Log  mean  distance  (uncorrected). 

The  same  corrected. 

True  mass  of  Neptune, 


170°  38'  48".7    4-8()i)8".  fi 
29    12  43.73  +  2811.4« 
73    14   37.6    +      29 .6« 
0    46   20 .92+        0.38/t 
.0463592  —  5236,a 
9562".27— 108".b« 
15424.797  — 0".838« 
1.2829251 + 179h  " 
1.2831223  -j-  179;t 

19700 


*f4ai- 


THE   ORBIT   OP   URANIJS. 


165 


Supposing  the  ninss  of  Noptunc  to  be  imccrtnin  by  oiu'-fiftietli  of  its  entire 
amount,  wliieli  is  quite  possible,  it  will  be  seen  the  longitude  of  the  mean  j)eii- 
helion  is  from  this  cause  uiicevtain  by  more  than  two  minutes,  the  mean  longitude 
of  Uranus  itself  by  nearly  a  minute,  and  the  mean  motion  by  nearly  two  seconds 
in  a  century. 

It  will  be  seen  that  the  logarithm  of  the  mean  distance  just  given  docs  not 
accurately  correspond  to  that  of  elements  I V'  jjIus  the  constant  term  of  iSv  X  0.-i343, 
as  it  should.  This  ditt'erenee  arises  from  the  rejection  of  the  terms  of  the  second 
order  in  hr),  which  can  not  affect  tlie  geocentric  longitude  of  the  planet  by  a  tenth 
of  a  second  for  a  number  of  centuries. 

It  is  to  be  remarked  that  these  mean  elements  arc  those  to  be  used  in  the 
general  theory  of  the  secular  varintion  of  the  planetary  orbits. 

Cviirhuhd  Theory  a/  l^raiiKx. 

The  elliptic  longitude  and  radius  vector  of  Uranus,  affected  by  the  secular  and 
long  period  perturbations  of  the  elements,  will  be  given  by  the  following  equations. 
Tut 

I,  =  iij  +  f(„ 
(j  =  l  —  7r,„ 
k  =  c,  +  tf>\ 

the  zeros  mdicating  elements  IV,  and  i7i,  fil;  and  &l  being  the  perturbations  of 
these  three  elements  just  given.     Then 
Elliptic  longitude  in  orbit  =  Z 

+  |2-]^e=-f    ^e'  I   |/.sinj/-Acos^J 

+ i  1  i!  ~  04 "' }  I  ^'' "  ^'''^'^ ''"  '^^  ~  ^^'''  ~  ^'"^ '"'  '^'A       . 

Nepcrian  logarithm  of  >•  =  n -|- I*?- -|-^,\,e' 

—  I  1  —  j^«''  I  I /'Cosr/  +  7t  sin^  J 

-  {  4  -  2I '"'  ]  1  ^''  ~  ''"'^  "^"^  ^^  +  ^'''' '"'  ^^  } 

24      May,  1873. 


'V 


T 


1  I 

I 
I 


M 


'0 

17 


li 


.     1  I 

I  1 


186  THE   ORBIT   OP    URANUS, 

~     24  {  ^''''  ~  "^'''''^  ""^  ^^  "^  ^'^'''''  ~  '*'^  *'"  ^^  } 

—     Ij J  I  (/.'  —  Gi7i^  4-  A')  cos  4j  +  (.4/.-7t  —  4M')  sin  iy  I 

In  computliifif  these  oxprcssions  it  will  bo  sufficient  for  scvorul  centuries  before 
or  after  1850  to  develop  h,  H;  and  <*/  to  tlieir  first  dimensions:  it  will,  liowever,  l»e 
more  convenient  to  correct  the  mean  anomaly  ij  for  the  perturbation  hi  before  obtain- 
infj  the  equation  of  the  centre;.  Dcvelopin<;  the  perturbations  of  7i  and  k  to  terms 
of  the  first  order,  we  have  for  the  effects  of  the  perturbations  of  those  elements: 


fk 


(('.c.l)  =  -(*2-^f',r)<7« 
{v.8.2)  =      (Jo  -  y^o' )  '^* 

(r.s.3)=      ^^c,,-Uc 

13  „„ 
(r.c.;J)  =  —  "4  <'"'  «« 

103 

(i-.c.4)  =  —  .j^V  '  ^« 
(p.f.O)  =       hn  -\-    (\,()7c 
(p...l)  =  _(l_|,„^),vA 

(p.c'.l)-=-(l-yv)^/^ 


(p.c.2)  =  —  2^0  '7'; 

17 

8 
17 


(p.«.3) 


e.'^h 


(p.c.3)=--  J^e,^7^ 

These  coefficients  for  p  must,  of  course,  bo  multiplied  by  the  modulus  0.434394 
to  reduce  the  perturbations  to  those  of  the  common  logarithm  of  the  radius  vector. 


*£i4." 


THE   ORBIT    OF   URANUS. 


187 


Among  the  elliiitir  terms  may  l)c>  iiuliulcil  the  effect  of  the  following  minute 
constants  introdjiced  by  tlie  peitnibutions. 

(ns.  2)  =_()".  144 

(w;.2)  =  +  0.l:{() 
0.434;}  (p.f.O)  =  -f  l\)V2  in  units  of  the  7th  place 
().4;34;j  (().«.  1 )  =  -j-  63  of  decimals. 

0.4343  (p.f.l)  =  + 73 
0.4343  (p.«.2)  =  4-5 
0.4343  (p.r.2)  =  4-4 

This  term  (p.c.O)  is  that  added  as  a  eonection  to  the  logarithm  of  the  mean 
distance. 

To  the  coefficients  (I'.sA),  (r.cA),  etc.,  arc  still  to  be  added  the  following  periodic 

terms : — 

1.  The  periodic  terms  due  to  tin;  action  of  Jupiter,  given  in  Chapter  V,  omitting 
the  terms  multiplied  by  T,  which  are  inchuled  in  the  perturbations  of  the 
elements. 

2.  The  periodic  terms  produced  by  Saturn,  including  those  terms  multiplied 
J)oth  by  7' and  by  sin  A.^  or  cos  A.,,  b»it  omitting  those  multiplied  by  7'  only  for 
tlie  same  reason  as  in  the  case  of  Jui)iter. 

3.  The  periodic  terms  produced  by  Neptune,  multiplied  by  the  factor  0.8(5294 
on  account  of  the  correction  to  tlio  mass  of  that  planet,  and  omitting  the  terms 
multiplied  by  hi,  he,  and  elg. 

4.  The  periodic  terms  multiplied  by  the  product  of  the  masses  of  Jupiter  and 
Saturn,  given  on  page  88,  omitting  the  terms  nndtiplied  by  the  sine  and  cosine  of 
N^  and  N-,  because  tiiey  are  replaced  by  tiie  terms  of  hi,  hh,  and  hi;  given  on  page 
183,  and  tabulated  in  the  columns  headed  (2)  on  page  184.  The  result  will  bo 
the  same  whetlier  we  employ  the  terms  of  (r.c.O),  (r.s.l),  etc.,  given  at  the  bottom 
of  page  88  and  tlie  top  of  page  89,  omitting  the  nundwrs  in  the  columns  2  on 
page  184  from  the  expressions  on  page  18G,  or  whether  we  include  the  latter  and 
omit  the  former. 

The  true  anomaly  of  Uranus  will  then  be : 

<jt)-\-hl-\-  (equation of  centre  from  elements  IV,  using  for  mean  anomaly  </o -\-  hi) 

-\- 1  (r.s.i)  sin  !ij  -}-  2  {i:c.i)  cos  ty. 

The  logarithm  of  the  radius  vector  will  be: 

log  ;•  in  elliptic  orbit  from  elements  TV. 
4"  -  (p-*-0  s'"  i'J  4"  '^  (p.c.i)  cos  iff 

care  being  taken  to  multiidy  the  coefficients  by  the  modulus  where  that  has  not 
already  been  done.     All  the  terms  in  Chapter  V  are  so  multiplied. 

To  pass  from  the  true  anomaly  to  the  true  longitude  we  must  investigate  tlic 
secular  motion  of  the  jdanes  of  the  orbit  and  of  the  ecliptic.  The  effi'ct  of  this 
motion  on  <}>,  0,  and  t  will  be  found  by  successive  api)roximations  from  the  formuke 


188 


THE   OIUUT   OF    U  II  A  NUS. 


(;U),  corrcctiiig  tlic  dutii  for  tlic  new  nuiss  of  Ncptiiiu'.     \Vc  shall  also  use  the 
saiuo  niotiou  of  the  (.•clii)tic  adoptud  on  j).  !).j.     \\C  liuvo  thus: 

til 


—  r.:>:\ 


^=  +  5.4:}4-0".;J87'. 
at 


lit 

lU 


=  —  5.17 
:=_4(;.7S-f  0.1'27'. 


Asa  first  approximation  wc  liave 

0  =  T  =  7:J^  14'     8"      —  31G!)".'i7' 
</)         =    0    4G    20.54+        '2.48  7' 

Substituting  these  values  in  (34)  and  integrating  we  find 

<^  =  <^,+  2".477'4-(r.l37'^ 
0  =  0„  —  3168  .42  7' -j- 3  .00  7'= 
T  =  Tu  —  31(i8  .70  7* -i-  3  .00  7'- 

For  tabulating  wo  shall  use,  instead  of  0  and  t,  the  distanro  of  the  perihelion 
from  the  ascending  node,  or  7t— t,  and  the  ^aluc  of  0  corrected  for  Struve's  pre- 
cession. Since  the  mean  motion  has  been  derived  without  nniking  any  distinction 
between  t  and  0,  it  will  be  necess;;iy  to  correct  the  motion  of  moan  anonndy  by 
the  difference  of  those  quantities.  We  tims  obtain  for  tiie  values  of  the  thvco 
principal  arguments:  — 

j/  =  220°  10'  10".35  +  lo42;)74".86  7'+« 
0)=    9;!      0   58.70+        31(58  .7(57'— 3.00 r^ 
0=    73    14     8. 00  4-        1856  .82'/'-]- 4.127'^ 

Tf  we  represent  all  tlic  inequalities  of  the  true  longitude  by  A/,  so  that  we  shall 
have  for  the  true  anomaly 

the  argument  of  latitude  will  be 

"=/+w. 
The  reduction  to  the  ecliptic  will  tlien  be 

It  =  —  (J)".37  +  O'.OIGT')  sin  2*^ 
the  true  longitude  on  the  ecliptic  referred  to  tlie  mean  equinox  of  date, 

and  the  sine  of  the  elliptic  latitude, 

sin  /3o  =  sin  <^„  sin  7*. 
The  perturbations  of  the  latitude  will  be 

(i.c.O)  +  (i.c.l)  cos  g  -\-  {h,a.\)  sin  g  -\-  etc. 


I  ■/*"■■*  f 


kJBJ 


TIIK   OUniT   or   U  II  ANUS. 


189 


Tl.o  lu-riodic  terms  of  (h.c.O),  (h.xA),  O'.c.l).  etc.,  arc  given  in  Chnpter  \  ,  on 
pa^cs  8G  ami  87,  und  nro  to  be  tuken  witliout  any  farther  niodiHeation  than  the 
niultiplieution  of  those  due  to  the  uetion  of  Neptune  by  the  factor  U.8Ua,  The 
constant,  seculur,  and  long  period  terms  arc 

/,.c.o  =+o".2()   —  ir.i'ir  —  o.()ii(V  +  <'*'^6,^x 

(6.*.1)  =  —0  .2'iT  —  O  .057"  +  ().!)75rV  -j-  O.'i'21/ix 
(h.c.l)  =  +'i  .47  T  4-  0  .127"  +  0.221^);  —  0.!)75ix 
{h.H.2)  =  -0  M  -  0  .01  7'  +  0.04fi,V  +  0.01  Ihx 
(i.f.'i)  =  _0  .01     -f-  0  .127'  +  0.011.V  —  0.0465x 

'rii(>  vahies  of  hr^  and  i^x  to  be  used  in  tliose  expressions  are  those  the  expressions 
for  whicli  are  given  on  page  97,  and  wliich  are  tabulated  in  the  last  two  columns 
of  the  table  on  page  184. 

The  following  tables  are  bused  on  the  elements  and  theory  laid  down  in  this 

chapter. 


Ul' 


190 


TUE    Oil  15  IT   OF    UUANUS. 


CHAPTER  IX. 


GENERAL  TA15LES  OP  UUAXUS. 


Emnmration  of  the  Quantifies  contained  in  the  several  Tahlca. 

The  first  six  tables  are  designed  to  give  the  values  of  tlic  three  arguments  of 
the  elliptic  motion,  g,  w,  and  0,  and  of  tlie  nine  arguments  of  the  tables  of  pertur- 
bations. The  argument  u  is,  however,  diminislied  by  3',  the  sums  of  the  constants 
added  to  the  perturbations  of  (i-.c.O)  to  make  these  quantities  positive,  and  0  by 
10",  the  constant  added  to  the  reduction  to  ecliptic.  The  expressions  for  the  argu- 
ments of  perturbations  are  as  follows,  the  mean  longitude  of  each  planet,  counted 
from  the  perihelion  of  Uranus,  being  represented  by  the  initial  letter  of  the  planet. 
All  these  arguments  are  expressed  in  units,  of  wliich  GOO  make  an  entire  circum- 
ference, so  that  each  unit  is  3G'.  The  time  t  is  counted  in  Julian  years  from  the 
fundanu  iitul  epoch, 

1850,  January  0,  Greenwich  mean  n.  on. 

Ar; 


1  = 

J— 

u 

=  219. 1 0( 

+  43.44028/ 

2  = 

S  — 

u 

=  577.34! 

1+13.22717/ 

3== 

U- 

X 

=    88.88-i 

t+  3.50035< 

4  = 

J- 

2S 

=  497.(5 

+    9.8445/ 

5=: 

3,^'- 

U- 

-    ,7=    79.8 

+    3.3825/ 

()  = 

46'  — 

nr- 

J=    57.1 

+  Kl.OlO/ 

7  = 

'2J  - 

3,S"  - 

-3r7=238.7 

+  18.()33/ 

8  = 

'iJ  — 

\S  - 

2f/=2()1.3 

+    5.4058/ 

y^ 

IS  - 

'2  J  - 

3^=130.9 

+  19.992/ 

Table  I  gives  the  corrections  which  must  be  applied  to  the  values  of  the  argu- 
ments at  any  time  during  the  nineteenth  century  to  reduce  them  to  the  corresponding 
time  in  any  preceding  or  following  century  between  the  Cliristian  era  and  the  year 
2300  Since  o  and  0  each  contains  a  term  proportional  to  the  square  of  the  time, 
the  correction  for  these  quantities  is  not  constant  during  each  century,  but  is  of 
the  form 

ij  +  cj'T* 

6)  and  J  being  constant  during  each  century,  and  T  being  the  fraction  of  the 
century  counted  from  its  beginning. 

Table  II  gives  the  value  of  7,  u  —  3',  0  —  10",  and  the  above  nine  argiiments  for 
(ircenwich  mean  noon  of  Jan.  0  of  each  leap  year  from  1752  to  1948,  and  for 
J  iniiary  —  1  of  the  years  1800  and  1900,  corresp'Muling  to  Decemb(>r  30  of  the  years 
1799   and   1899.     Tlu:   corrections  for  the  perturbations  of  long  period  are  not 


k3 


THE    ORBIT   OP   URANUS. 


191 


applied  in  this  tabic.  The  numbers  at  the  bottom  of  this  table,  in  the  line  AfJi, 
sliow  the  variation  of  the  corresponding  quantity  in  I'iO  days,  for  the  epoch  1850.0. 
In  tiie  line  "Factor  7'"  is  given  the  change  of  tliis  variation  in  a  century,  while 
Alii  is  the  second  difference  for  intervals  of  120  days,  liy  means  of  tliese  num- 
bers, when  the  arguments  are  computed  for  any  date,  their  values  for  other  dates 
at  intervals  of  120  days  may  be  found  by  successive  addition. 

Table  III  gives  the  motion  of  tiie  several  arguments  between  the  epochs  of  the 
preceding  table  and  the  zero  day  of  each  montii  in  the  course  of  a  four-year  cycle. 
The  variable  motions,  w  and  0,  correspond  to  the  epoch  1850,  and  rigorously  they 
each  require  a  correction  for  any  other  four-year  cycle  tlian  that  between  1848  and 
1852.  But,  owing  to  tlie  small  inclination  of  the  orbit  of  Uranus  it  is  not  neces- 
sary that  either  «  or  0  should  be  exact,  if  only  their  sum  is  exact.  Tlie  column 
0'  of  this  table,  therefore,  gives  the  correction  wiiicli  nuist  be  applied  to  the  motion 
of  0  at  the  end  of  a  century  (1050)  in  order  that,  being  applied  to  0  alone,  (j -)-  0 
may  be  exact.  Tliis  correction  is,  in  fact,  tiiat  for  the  secular  variation  of  the 
precession. 

Tables  IV  and  V  give  the  motion  of  the  arguments  for  days  and  hours.  The 
motion  for  hours  is,  liowever,  not  necessary  in  the  case  of  any  argument  but  (j,  as 
all  the  others  can  be  readily  enougli  interpolated  to  fractions  of  a  day. 

Table  VI  gives  the  corrections  to  the  arguments  on  account  of  the  terms  of  long 
period  from  1000  to  22()(T.  The  terms  in  question  are,  in  the  case  of  Jupiter,  tlic 
great  ineciuality  produced  by  the  action  of  Saturn,  in  the  case  of  Neptune  tlie 
great  inequality  produced  by  Uranus,  and,  in  the  case  of  Uranus,  the  iiu'qualitics 
in  the  mean  longitude  tabulated  in  the  preceding  chapter.  The  numerical  expres- 
sions are 

U=  0.5:J5  sin  (110°  21'  +  40^  45'  20"  T) 

hu=n 

The  corrections  to  the  several  arguments  are 


b  arg.  1 


=       (7 


U—H 


b  arg.  2 


-rV 

S  ar?'  3  =       (V 


,").V=1.75,y 


Xo  correction  to  the  mean  longitude  of  Saturn  is  applied,  all  its  inequalities  being 
taken  account  of  in  the  terms  of  the  second  order. 

The  corrections,  expressed  in  seconds,  have  been  reduced  to  units  of  the  argu- 
ment by  dividing  them  by  21(iO". 

Outside  the  limits  of  the  table  these  corrections  must  be  computed  from  their 
formuhe. 

Table  VII  gives  the  equation  of  the  centre,  and  the  elliptic  part  of  the  logarithm 
of  the  radius  vector.  No  constant  is  ajiplied  to  the  former,  but  the  latter  is  dimi- 
nished by  .0()0:M00,  the  sum  of  the  constants  added  to  (ji.r.O)  in  Tables  VIII,  IX, 
X   and  XVII. 


192 


THE   ORBIT   OF   URANUS. 


The  formulae  for  the  Tables  are 

Equation  of  centre  =        193iV2".()()  sin 
-j-       5G7  .24  sin 
4-        23  .05  sin  3j 
+  1  .07  sin  4<j 

-\-  0  .05  sin  b<j 

Elliptic  log.  r  =        1.2833435 

—  .()()03400 

—  .0203618  cos   g 

—  .0007 1G5 cos 2y 

—  .0000318  cos  3i/ 

—  .0000016  cos  4</. 

Table  VIII  gives  the  coefficients  (y.c.O),  (I'.c.l),  etc.,  for  tlio  perturbations  of  the 
longitude  and  logarithm  of  radius  vector  produced  by  the  action  of  Jupiter.  They 
arc  computed  from  the  periodic  terms  of  the  fornuilic  on  page  83,  with  tiie  addi- 
tion of  the  following  constants  to  make  all  the  numbers  of  the  table  positive : 

Constant  of  (e'.c.O)  =  55". 
(r.s.l)=    6. 
(r.c.l)=7^    4. 
{v^  2)  =    0  .20 
(W.C.2)  =    0  .20 
(p.c.O)  =  1200 
(p.s.l)=    150 
(p.c.l)=    100 
(p.,'(.2)  =      10 
(p.c.2)=      10 

Table  IX  gives  the  periodic  part  of  the  coefficients  due  to  the  action  of  Saturn, 
taken  without  change  from  the  expressions  on  page  84,  together  with  the  secular 
variations,  the  latter  including  only  the  terms  of  (t-.s.l),  (t-.c.l),  (i-.s.2),  and  (r.c.2), 
which  are  multiplied  by  T  and  by  sin  Jj  or  cos  A.^.  The  coefficients  of  T  are  given 
hi  the  columns  Sec.  Var.  and  each  number  is  increased  by  the  constant  1".50  to 
make  it  positive.  The  term  — 0".06rsin  A.^  in  {r.c.O)  is  omitted  entirely,  as  it  will 
not  amount  to  a  tenth  of  a  second  until  after  the  year  2000.     Tiie  constant  terms 

added  to  the  quantities  of  tiicse  tables  to  make  all  tl-.e  numbers  positive,  are : 

'/         // 
Constant  of  (r.c.O)  =    30. 

(r..9.1)  =  150.  +  1.50r 
(I'.c.l)  =  150.  +  i.5or 
(M.2)  =  130.  -I-  \MT 
{r.c.2)  =  130.  +  1.507' 
(V..«.3)=      8. 
(t\c.3)=      6. 
(r...4)  =1. 
((•.C.4)  =1, 


THE   ORBIT   OP   URANUS. 


193 


Constant  of  (p.c.O)  =  800 
(p.8.1)  =  1500 
(p.c.l)  =  1500 
(p.s.2j  =  400 
(p.c.2)  =  400 
(p.s.3)  =  100 
(p.c.3)=    100 

Tabic  X  gives  the  coefficients  produced  by  the  action  of  Neptune,  computed 
from  the  periodic  tonus  on  pages  86  and  86  without  any  other  change  than  the 
multiplication  of  all  the  numbers  by  the  factor  0.863  to  reduce  them  to  the  new 
mass  of  Neptune.     The  constants  added  to  the  several  quantities,  are 


Constant  of  (I'.c.O)  = 

92.85 

(v.s.l)  = 

20.00 

(v.c.l)  = 

31.00 

(v.s.2)  = 

5.00 

{V.C.2)  = 

5.00 

(r..s.3)  = 

1.00 

(r.0.3)  = 

1.00 

(r..s-.4)  =  ■ 

-    1.00 

(r.6'.4)  = 

-    1.00 

Constant  of  (p.c.O)  = 

400 

(p.«.lj  = 

200 

(p.c-.l)  = 

200 

(p.s.2}  = 

40 

(p.c.2)  = 

40 

Tal)les  XI  to  XVI  give  tlie  terms  of  the  second  order  and  of  short  period  which 
contain  tlie  products  of  the  masses  of  Jupiter  and  Saturn,  which,  with  the  constants 
added  to  the  niunbers  of  the  several  tables,  are  as  follows: 

{p.c.O)  =  +  0".m  sin  A,  +  0".51  cos  A, ;    Table  XII ;    const  =  0".6n 

—  0.01  sin  .1,-1- 0.05  cos  ^;; 

—  0.35  sin  .1,  —  1  .30  cos  .4,; 

—  0  .05  sin  .1,, -f- 0  .03  cos  .4,; 

Sum  of  constants  added  to  these  tables 

(r.«.l)  =  +  0-.2fi  sin  ^,  -f-  0".27  cos  A,; 

—  0.04  sin /I, —  0.17  cos  yl,; 

-l-0.08sin  y1„-|-0.03cos^1.,; 

-0.02  sin  A,  -f- 0.08  cos  .4;; 

-I- 0  .30  siiwl,  —  0  ,58  cos  .4,; 

—  0.04  sin  A,; 

25       Juno,  1373. 


XUI; 

=  0  .05 

XIV; 

=  0  .05 

XV; 

=  1  .35 

XVI; 

=  0.10 

ibles 

2.15 

Table  XI; 

const  =  0".40 

XII; 

=  0  .20 

XIII; 

=  0.10 

XIV; 

=  0.10 

XV; 

0  .75 

XVI; 

0.10 

li: 


194 


THE   ORBIT   OF   URAXUS. 


Table  XI; 

const 

=  0".40 

XII; 

0  .20 

XllI; 

0.10 

XIV; 

0.10 

XV; 

0.75 

XVI; 

O.IO 

in  these  ti 

ibles 

1.65 

(V.C.I)  =  -I-  0".06  sin  At  -  0".27  cos  A,; 
+  0.18sin^, -fO.Olcos^; 

—  0  .03  sin  A^  +  0  .08  cos  A,r, 

—  0. 0-2  sin  it, -f  0.09  cos  A; 

—  0  .44  sin  A^  —  0  .61  cos  A^; 

—  0.1 0  sin  4, +  0.03  cos  J,; 
Snm  of  constants  added  to  (v.s.l)  and  (c.c.l)  in  these  tables 

The  term  of  (p.c.O) 

1 1  sin  A^  —  3  cos  A^ 
is  omitted  from  the  tables  entirely. 

Tables  XVIIa  and  XVII i  give  the  constant,  secular,  and  long-period  terms  of 
{r.fi.l)  (v.c.l),  computed  from  the  fornuilic  p.  186,  with  the  following  additions: 

1.  The  constant  terms  introduced  by  the  ])erturbations,  given  on  p.  187. 

2.  Tlie  negatives  of  the  constants  added  to  the  tables  VII  to  XVI  inclusive  to 
make  the  numbers  of  those  tables  positive.     The  values  of  these  terms  are 


Pert.  Const. 

Tobies  Till  t( 

)  XVI. 

{l)-(2) 

(v.s.l) 

0 

177".65 

+  1' 

.50  7' 

_177".65  — 1".507' 

(V.C.I) 

0 

186  .65 

+  1 

.50  r 

—  186.65  —  1.50  7' 

(r...2) 

—  0".14 

135  .20 

+  1 

.50  7' 

—  135.34  —  1  ..507' 

(r.c.2) 

+  0.13 

135  .20 

+  1 

.507' 

—  135.07  —  1.507' 

(r..s.3) 

0 

9  .00 

—      9 .00 

(v.c.-i) 

0 

7  .00 

—      7.00 

(p.c.O) 

[+1972] 

—  1000 

+  1000 

(p.«.l) 

+     63 

1850 

—  1787 

(pel) 

+     73 

1800 

—  1727 

(P....2.) 

+       '"> 

450 

—    445 

(p...2) 

+       i 

450 

—    446 

The  perturbation  constant  of  (p.c.O),  being  added  to  log  a  in  forming  tiie  elliptic 
radius  vector,  is  not  included  in  this  table. 

Table  XVIII  gives  the  reduction  to  the  ecliptic 

—  9".37  sin  2>,. 

The  constant  10"  is  added  to  make  the  numbers  always  positive,  wliich  constant 
has  been  already  subtracted  from  (). 

Table  XIX  gives  the  principal  term  of  the  latitude 

4(i'  20".54  X  sin  ». 

'fable  XX  gives  the  coefficients  {h..s.l)  and  (h.c.])  for  the  jjerturbations  of  the 
latitude  produced  by  Jui)iter.     They  are  given  by  the  formuhc 

(-/>.«.!)=:  0'.65  cos  (,/— r+ 40^) 
(h.c.l)  =  0  .65  sin  (./—  U -\-  40^) 

The  ronstnnt  0''.70  is  added  to  make  all  the  numbers  of  tlie  tabl(>  positive. 


THE    OUBIT   OF   URANUS. 


195 


Table  XXI  gives  the  conospouding  coefficients  for  tlie  action  of  Saturn,  com- 
puted from  the  expressions  on  p.  87  witli  the  addition  of  the  following  constants. 

Const,  of  {h.c.O)  =  0".lO 
(i..v.l).-=;}..3() 
(i.e.  1)  =  3.10 
(i.«.2)  =  0  .20 
(6.C-.2)  =  0  .20 

Tabic  XXII  gives  the  coefficients  for  the  action  of  Neptune  from  the  formuhe 
on  p.  87,  all  the  numbers  l)eing  multiplied  by  the  factor  ().8();3  to  reduce  them  to 
the  adopted  mass  of  Neptune.     The  following  constants  are  added : 

To  (i.f.O)  ....  0".06 

(i.s.l)  1  .00 

{h.c.l)  1  .20 

(/>.s.2)  0 .20 

(A.r.2)  0.20 

Table  XXIII  gives  the  secular  and  long-period  terms  for  various  epochs  com- 
puted from  the  formula?  of  p.  189.  The  sums  of  the  several  constants  added  in 
the  three  preceding  tables  are  here  subtracted  again  so  that  these  expressions 
become 

h.rJ)   —  2c=       0".10-0".12r—     .01b^>7  +  .046,^x 
{/t.f.l)  ~^c^  —  o  .00  —  0  .227'  —  0'.orj7'-   +  .975^>7  +  .221Sx 


(A.c.l)  _  vc  =  _  5  .00  +  2  .47T+  0  .12'/'^  +  .22bV 
(/>.«.2)  —  2c  =  —  0  .40  —  0  .01  7' 4-  .046,V  +  -Ollf^K 
(/AC. 2)  —  2c  =  —  0  .41  +  0  .127'+     .OIU*?  —  M68x 


.9'iobx 


Prcccjyfs  for  the  nse  of  tli(  Talks. 

Express  the  date  for  which  the  position  of  Uranus  is  required  in  years,  months, 
days,  and  hours  of  Greenwich  mean  time,  according  to  the  Julian  Calendar  if  the 
date  is  earlier  than  1500,  according  to  the  Gregorian  Calendar  if  it  is  later  than 
IfiOO,  and  according  to  either  calendar  between  these  epochs. 

Enter  Tabj  I  with  the  beginning  of  the  century,  and  take  out  the  values  of  g, 
«,  J,  6,  0',  and  arguments  1  to  9.  Multiply  w'  and  (/  by  the  fraction  of  a  century 
corresponding  to  the  date,  and  write  the  products  with  their  proper  algebraic  signs 
under  o  and  0.  If  the  calendar  is  the  Julian,  the  century  marked  ,7"  must  be  takcMi, 
and  if  the  Gregorian,  that  marked  G.  Between  the  dates  1752  and  1951  it  is  not 
necessary  to  enter  Table  I  at  all. 

If  Table  I  was  not  entered,  enter  Table  II  with  the  yenv,  (U*  the  first  preceding 
year  found  therein.  If  Table  I  was  entered,  enter  Table  II  between  the  year 
1800  and  1896  as  if  the  niimber  of  the  c(>ntury  wer<>  changcul  to  18.  Take  out 
th(!  values  of  <f,  o,  0,  and  the  arguments,  and  write  them  under  the  corresponding 
quantities  from  Table  I. 

Enter  Table  III  with  the  excess  of  the  actual  year  over  that  with  which  Table  II 
was  entered,  and  with  the  month.  Write  the  corresponding  values  of  //,  w,  0, 
and  the  arguments  under  the  previous  values.     Multiply  0'  by  the  fraction  of  a 


196 


THK   Oil  HIT    OF    URANUS. 


\ 


ccn  Mry  after  1850,  corresponding  to  the  date  with  which  Tahle  II  was  entered, 
and  '.vrite  the  product  under  (),  or  add  it  to  it  in  writing  {).  11'  Table  11  was 
entered  witii  a  date  before  1850,  this  product  is  negative. 

Enter  Table  IV  with  the  day  of  tiie  month  and  write  down  the  corresponding 
values  of  g,  u,  etc.,  inidcir  the  former  values. 

If  the  date  does  not  corresjxmd  to  Greenwich  mean  noon,  the  motion  of  y  for 
the  hours  must  be  computed  from  Table  V,  and  the  other  quantities  must  be 
interpolated  to  the  fraction  of  a  day  in  entering  Table  IV. 

Enter  Table  ^'I  witli  the  year,  find  by  interpolation  tlie  values  of  //,  and  argu- 
ments 1,  2,  and  3,  corresponding  to  the  date,  and  write  them  under  the  former 
values. 

Add  up  all  the  partial  values  of  g,  o,  0,  and  the  arguments,  attending  to  the 
algebraic  signs  of  tin;  products.  Snbtrac^t  from  tlie  ar<j;uments  as  many  times  GOO 
as  ])o9sible,  and  the  results  will  be  the  final  values  of  those  quantities. 

Enter  Table  VII  with  g  as  the  argument,  the  seconds  being  first  reduced  to  frac- 
tions of  a  minute,  and  interpolate  the  quantities  £'and  log  r.  When  g  exceeds  180° 
the  former  (piantity  is  to  receive  the  negative  sign;  the  latter  is  always  positive. 

Enter  Tables  VIII  to  XVI  inclusive  with  tbeir  respective  arguments,  and  take 
out  the  values  of  the  quantities  (r.r.O),  (r..s.l\  (v.c.l),  etc.,  (p.c.O),  (p.».l),  etc.,  so 
far  as  they  are  found  in  the  tid)les,  wvitiiig  the  qtiantities  having  the  same  desig- 
nati(m  under  each  other.  In  Table  IX  the  quantities  Sec.  ^'ar.  must  be  nndtiplied 
by  th(>  centuries  and  fraction  of  a  centiny  of  the  actual  date  after  1850,  and  the 
l)rodiHt  must  be  included  witli  the  corresponding  quantities,  (r.'i.l),  (r.c.l),  etc. 
Before  1850  this  product  will  always  be  negative;  aftmward  always  positive.  All 
the  quantities  taken  from  these  tables  are  positive  except  (f.«.4)  and  (r.c.4)  in 
Table  IX,  which  are  negative. 

Add  up  all  the  partial  values  of  (r.c.O),  (r..'».l),  etc.,  thtis  obtained  from  Tables 
VIII  to  XVI,  and  from  their  sum  take  the  corresponding  quantities  obtained  from 
Tabl<>  XVll  by  interpolathig  to  the  date.  The  rc^quired  quantities  arc  all  given 
in  Table  XVII A  ;  Table  XVlIa  being  only  an  expansion  of  a  i)art  of  XVIli  for 
the  present  century.  The  final  values  of  (r.s.l),  (r.c.l),  (r.s.2),  etc.,  (p.s.l),  (p.*".!), 
etc.,  thus  obtained  are  to  be  multiplied  by  the  sines  and  cosines  of  the  correspond- 
ing multiples  of  g,  in  doing  which  four  place  logaritlmis  are  sufficient  if  the  coni- 
jMitation  is  carefully  mad(\  Tlie  products  are  then  all  added  togetlii^r,  and  to  f/,  w,  7s', 
and  (r.c.O);  in  the;  case  of  r,  and  to  log.  r,  (p.c.O)  in  the  case  of  p.  That  is,  we 
are  to  form  the  expressions; 

u==</-|-u-f-£+ (t'.c.O) --)- (r..f.l)  sin    r/ -f- (r.c.l)  cos   g 

-f  (r..s.2)  sin  2/  +  (<'.c.2)  cos  2j 
-f-         etc.         -\-         etc. 

log  r  —         log  r  (from  Table  VII)  +  (p.c.O) 
4- {p.«.l)sin    g)       +(p.c.l)cos   g 
-f  (p.«.2)  sin  2;/)       -f  (p.c.2)  cos  1g 
-|- (p.s.3)  sin  3y)      4"  (P'^'-'^)^'^'*%' 


TlIK   OUBIT   OP   URANUS. 


197 


u  will  then  be  the  true  argument  of  latitude,  and  log  r  the  loguritlnn  of  the  radius 
vector  with  seven  places  of  decimals. 

Under  n  write  0;  enter  Table  XVIII  with  the  argument  m  and  take  out  the 
reduction  to  the  ecliptic.  Add  it  to  u  and  0,  and  the  sum  of  the  three  quantities 
will  be  the  heliocentric  longitude  of  Uranus  referred  to  the  mean  equinox  and 
ecliptic  of  the  date.  Applying  nutation  the  longitude  will  be  reduced  to  the 
true  equinox. 

Enter  Table  XIX  with  u  as  the  argument,  or,  when  u  exceeds  180°,  with 
u  —  180°,  and  take  out  the  principal  term  of  the  latitude,  which  will  be  positive 
when  u  is  less  than  180°,  and  negative  when  it  is  greater. 

Enter  Tables  XX,  XXI,  XXII,  and  XXIII  with  their  respective  arguments, 
the  argument  for  the  last  being  the  date,  and  add  up  the  various  (piantities  having 
the  same  designation,  noticing  that  in  the  first  three  tables  all  the  (piantities  are 
positive,  while  in  the  last  they  are  all  negative  except  (i.c.O).  Then  form  the 
expression, 

{h.c.O)  +  {h.s.1)  sin  g  +  (h.c.l)  cos  g  +  {h.M.2)  sin  2j  +  a>.c.2)  cos  Qg, 
and  add  it  to  the  principal  term  of  the  latitude,  with  regard  to  the  algebraic  signs. 
Tlie  sum  will  be  the  heliocentric  latitude  of  Uranus  above  the  ecliptic  of  the  date. 
"SViien  an  ephemeris  of  Uranus  is  to  be  conq)uted  for  a  series  of  years,  some 
modifications  may  be  introduced,  which  will  save  the  computer  labor.  In  the  first 
place  an  ecpiidistant  series  of  d.ites  being  selected  for  computation,  it  will  be  suffi- 
cient to  compute  g,  u,  0,  and  the  arguments  for  every  sixth,  eighth,  or  tentli  date, 
and  to  fill  in  the  arguments  for  the  intermediate  dates  by  adding  the  nearly  con- 
stant ditterences  corresponding  to  the  adopted  intervals.  The  agreement  of  the 
numbers  thus  obtained  for  the  last  date  with  those  found  by  the  original  computa- 
tion will  prove  the  whole  process.  This  interval  may  be  as  great  as  l'>()  days 
without  detracting  from  the  accuracy  with  which  the  places  for  the  immediate 
dates  can  be  interpolated,  and  the  diff'erences  for  this  interval  may  be  deduced 
from  the  numbers  at  the  bottom  of  Table  II.  If  these  numbers  are  used  without 
change  the  values  of  w  and  0  for  the  last  date  muy  not  always  come  out  right. 
But  these  errors,  if  less  than  a  second,  will  be  of  no  importance  if  the  one  quan- 
tity comes  out  as  much  too  great  as  the  other  is  too  small,  and  they  may  be  avoided 
entirely  by  making  a  small  cliange  in  the  constant  difterence  to  be  added. 

Tables  XI  to  XVI,  inclusive,  need  be  entered  only  for  every  third  or  fourth  date, 
and  the  sums  of  the  quantities  can  be  then  interpolated  to  every  date,  and  added 
up  with  the  corresponding  quantities  from  the  other  tables. 

Again,  it  will  be  found  convenient  to  compute  the  sum  of  the  small  terms 
(y.s.3)  sin  %  +  0'<^--^)  ^^^  '^.Z  +  ("•^''•*)  ^'"  "^'^  +  {r.cA)  cos  4r/,  as  well  as  the  corre- 
sponding terms  of  the  radius  vector,  and  all  the  terms  of  the  latitude,  not  for  the 
dates  adopted,  but  for  every  fourth  entire  degree  of  g.  Having  a  series  of  values 
computed  in  this  way,  the  sum  can  be  interpolated  to  the  value  of  g  corresponding 
to  the  date.  To  facilitate  the  formation  of  the  smaller  products  for  entire  degrees 
of  //,  a  table  of  products  of  numbers  by  the  sine  and  cosine  of  every  degre(>  is 
appended  to  these  tables,  by  which  the  products  in  question  can  be  formed  at  sight 


ji 


198 


THE    OUBIT   OF   URANUS. 


wlioncvor  the  coefficient  to  bo  nmltiplicd  is  less  than  3'2".  Tiie  Viihies  of  these 
coofficieiits,  (f.«.;3),  (r.r.'A),  etc.,  corresjioiuling  to  the  entire  dej^ree.s  of  //,  may  ho 
eitlier  foiinod  by  interpohition  at  sight  from  tiiose  corresponding  to  tiie  dates  of 
computation,  or  the  vahies  of  the  arguments  2  and  !i  corresponding  to  tin; 
required  degrees  of  ij  nniy  be  computed,  and  the  values  of  {r.x.'.i),  etc.,  correspond- 
ing to  these  values  of  the  arguments  may  be  taken  from  Tables  IX  and  X,  while 
Table  XVII  must  bo  entered  with  tlie  corresponding  dates. 

If  tlic  lieliocentric  ephomeris  is  coinput(;d  for  ten  years  at  a  time,  tlio  last  of 
those  modifications  in  the  mode  of  computation  will  greatly  facilitate  the  computa- 
tion of  tiie  smaUer  terms.  We  first  lind  tlie  date,  and  the  values  of  arguments  1, 
2,  and  3,  to  one  jdace  of  decimals,  for  some  entire  degree  of  y  preceding  that  wiiicli 
corresponds  to  the  first  date,  and  tlien  find  the  dates  and  the  values  of  the  arguments 
corresponding  to  successive  values  of  (/,  differing  by  2"^  or  4°,  until  we  pass  the  last 
date  of  computation.  We  then  take  out  the  values  of  (?\«.3),  (r.c.3),  ((•.«.4),  ((•.c.4), 
(p.«.3),  (p.c.3),  (/(.c.O),  (ft.fl.l),  {h.c.l),  {l)..i.2),  and  (/>.c.2),  with  these  values  of  the 
dates  and  iirguments,  form  their  ])roduets  by  the  sines  and  cosines  of  th(>  corre- 
sponding multij)les  of  ij  by  means  of  tiie  supplementary  tables,  and  add  tiie  proper 
products  together  so  as  to  fori:;  three  small  tables  witli  tj  as  the  argument.  These 
terms  are  then  iuteipulated  to  the  values  of  (j  corresponding  to  tlie  original  dates 
of  computation. 

As  a  iinit  e.\ampl(<  of  the  use  of  the  Tables  we  will  compute  the  heliocentric 
co-ordinates  of  Uranus  for  Greenwich  mean  noon  of  tlie  date  1753,  Dec.  3.  In 
computing  the  argninents  we  sliall  make  use  of  Talde  I,  tiiongh  it  is  not  necessary 
to  do  so.     The  computation  of  the  arguments  is  as  follows: 


u 

u 

0 

7 

Arg.  1 

Tabic  1,  1700 
PriHliicl  hv  a.5:!02 
Tablu  Il,"ls52 

111,  Y.  1,  Dec. 
]V,  3  iliivs 
VI,  175:!  02 

o              / 

2-.ll     31 

n 

7. 

0.73 
43. 3S 

(>.70 
J2.30 

30.48 

0 

:!50 

04 
0 

/           // 

7       5.33 

+3.24 

50       2.03 

1        0.73 

0.2(1 

o              t 

350     2'J 

73     14 
0       0 

// 

11.4 

4  4 

450.002 

300.010 

83.252 

0.357 

0.507 

22S     44 
S     12 
0       2 
0       0 

108     30 

35. 1 1 

35.50 

0.15 

1753,  Dec.  3 

94 

7     11.50 

72     44 

17.87 

240.218 

Arg.  2 

3 

4 

5 

0 

7 

8 

9 

401 

177 

38 

0 

Tabic     I,  1700 
II,  18,52 
HI,  y.  1,  Dec. 
1            IV,  3  (lavs 
'            VI,  1753.0 

477.310 

3.785 

25.340 

0.100 

—0.014 

500.548 

240.075 

05.880 

G.700 

.020 

+  .024 

210 

517 

10 

0 

202 

S7 

0 

0 

130 

00 

32 

0 

537 

270 

30 

0 

59.4 

272.1 

10.3 

0.0 

175.3,  Dec.  3 

352.  CI  7 

1.52 

355 

201 

240 

341.8 

10 

THE   OUBIT   OF   TJRAXUS. 


li)9 


(vr.O)      (r.i.l)       (I'.c.l)       (i'.«.2)  (v.c.2)  (.'.s.S)      (r.0.3)     (r.xA)     (r.r.i) 

,,,,/,               "  '.  "                "                "               " 

0.37  0.31 

CO.'M  i-M.H'i  1.42         11.07         0.32         1.04 

0.04  5.3a  0.70           1.1'J     — O.O;:     —1.04 


Table  VIII 

83.58 

9.97 

6.0.1 

IX 

21.32 

22.24 

223.49 

X 

71. OS 

10.02 

is. 30 

XI 

.13 

.47 

XII 

.13 

.3(1 

.10 

XIII 

.or, 

11 

.01 

XIV 

.01 

.04 

.03 

XV 

2.38 

1.40 

1.12 

XVI 

.12 

.09 

.11 

V 

178. <;» 

53.31! 

250.24 

Table  XVII 

— 2!)2.r),5 

—140.91 

—230.19 

-j-109.35 

(;7.!I3       244.58         2.12         12.2(1 
—140.73  —131.03     —9.41       — (i.M5 


—  72.80  -1-113.55     —7.29       -|-5.41 


Tabic  VIII 

IX 

X 

XVII 


(l>.''0) 
251 

1080 
320 

1003 
1104 

2707 


(p.x.l) 
188 
741 
259 

118S 
—1321 

—  133 


(P.''.l) 

32 

300 

222 


(p.s.2)  (p.*-.2) 


021 
21 


014 
—500 

+  114' 


042 
—412 


-f230 


507 
42 

039 
—300 

+279 


—0.03 
{,,....  3) 

154 

r  I 

—08 
+50 


+0.00 
(,,.c.3) 

111 


HI 

—04 


l0(r(l'..S.l)—  2.37H7 

lipjr     sill      +9.2004 

luhr(p..s.l)  -2.124 
o  / 

30 


108 

91 

1 


7 
0 
2 

—  1 
1 


9 

U 

K 

(•!.<'.0) 

(r.,s.l)siii  <j 
(i'.c.l)  ens  (J 
(i'..s.2)  sin  23 
(y.e.2)  cos  2^ 
(i).s.3)  sin  3^ 
{v.c.'ii)  cos  31/ 
(U..S.4)  sin  ig 
(v.r.i)  cos4</ 

u 

e 

li 
Loiiir.  mean  Ki\.  330     25 
Nntalion 


203 
72 


40 
44 


lotr(r.c 
loj;    CIIS 

log(|,.C. 

II 

30.18 
11.50 
40.21 
58.08 
-47.00 
47.15 
2H.43 
4  4. 55 

-  4.13 

-  4.40 

0.4T 
0.41 

57.42 

17.87 

7.95 

23.24 

+  0.51 


.1)  +  2.0388 
f/  _  0.0012 
1)   +2.057 

Tabic 


l»g(i'.«.2)—  1.8021 
loj?  sin  2-/  —0.5010 
log  (p.«.2)+ 2.302 


Ii>j?(ii.<'.2  +  2. 
lojr  cos  2,7  +  0. 
lui;0,.r.2)  +  2. 


0,552 
0041 
440 


XX         ".00 

X.\I  3.01 

XXII      _  .84 

4.81  7  .80         6^59' 

XXIII       —4.28       — 0  .88     —0.43 

+  0 .53        +  0  .  08     +  (f.  1 0" 


(h.r.\)  (6.,s.2)       (/».c.2) 
0".02 

5  .88  0.41 

1  .30  0.18 


0.17 
_0.I7 

0.34' 
— 0.51_ 
""—0.17 


Long,  true  Kq.  330     25     20.75 


Tabic 
(I'.f 

(p..S 

((>.(• 
(p..s- 

(p.. 

(p..s 

(p.r 

log 


VII 

<•) 

1 )  **'"  .'/ 


.  1  )  COS  (I 

.2)  .sin  2;/ 
.2)  cos  2.7 
.3)  sin  3f/ 
3)  cos  3;/ 


I<ig  )• 
1.3023222 
2707 

—  2(! 
—112 

—  00 
+257 

+  ;<i 

—  14 


1.320035 


l^^   Table  XIX  —0 
(l).c.O)   Table  XXI 
XXII 
XXIII 

(li.xA)  sin  g 
{h.cA)  I'OHg 
(fix. 2)  sill  2.7 
(fc.f.2)  cos23 
Latitude        —  0 


40 


40 


3. 
0 
0 
0 

+  " 
— 0 
— 0 
— 0 
4 


"73 
.10 
.09 
.20 
.11 
.00 
.00 
.10 
.35 


As  a  second  example  wc  will  take  the  comptitation  of  an  ophemoris  for  tiie  years 
187G  and  1877.  AVe  take  as  the  extreme  dates  1875,  December  15,  and  1878,  April 
;},  between  whii^h  are  seven  intervals  of  I'iO  days  eneb,  wbich  wi-  adopt  as  those 
of  compulation.     We  lirst  form  the  uryiuueuts  for  the  e.xtn'uie  dates  as  follows : 


■f'f 


200 


1  a  i:   O  11  D  1  T  OF   U  U  A  N  u  s. 


1.    For  1875,  Dec.  1,'>  —  1875.90. 

9 

u 

0 

Arg.  1 

Table  11,  1872 

III,  3  Y.  Doc. 

IV,  15ihiy« 
VI,  1875.90 

a              1              It 

314     25     6S.70 

IG     4(!     CC.7C 

0     10     33.. 50 

2.20 

0 

95 
0 

1 

9     3 
2 

If 

5.04 
4.00 
1.30 

0                   /                   1/ 

73     20     40.0 

0       1     12.7 

0.7 

7 
> 

0 

574.815 

170.072 

1.784 

+    0.420 

147.097 

For  1875,  IH'o.  15 

331     23       5.10 

95     11     41.00 

73     22       0.15 

Arg.  2 

3 

4 

5 

C 

7 

8              9 

Tubloll,  1872 

III,  3  Y.  Doc. 

IV,  15  (liiv.s 
VI,  1875.90 

208.328 

51.785 

0.543 

—     1 

320.055 

105.888 

13.705 

0.144 

+      2 

114 

39 
0 

154 

13 

0 

423 

05 

1 

49 

73 

1 

380.3 

21.1 

0.2 

577 

78 

1 

For  1875,  Doc.  15 

179.739 

l5;j 

107 

489 

123 

401.0 

50 

11.   For  1878,  April  3  =  1878.20. 

9 

M 

0 

Arg.  1 

148.570 

97.043 

.357 

.421 

Table  ir,  1870 

III,  2  Y.  April 

IV,  3(1 

VI,  1878.26 

o           1           n 
331     31     18.70 
9     37     53.02 
2       0.70 

2.02 

0 

95 
0 

/         // 
11     42.33 
1     11.23 
0.20 

0           /                ;/ 

73     22       1.(1 

0       0     41.7 

0.1 

5 
5 

For  1878,  April  3 

341     14     21.04 

95     12     53.82 

73     22     42.93 

240.997 

Arg.  2 

3 

4 

5 

G 

7 

8              9 

Table  II,  1870 

III,  2  Y.  April 

IV,  ;!(1 

VI,  1878.20 

321.237 

29.732 

.109 

—  .001 

351.077 

179.889 

7.808 

.029 

4-     .002 

153 

22 

0 

108 
7 
0 

175 

489 

37 

0 

123 

42 

0 

401.9         57 

12.1         45 

0           0 

i 
1 

For  1878,  April  3 

187.788 

175 

520        105 

414.0       102 

We  now  fill  in  the  values  of  (7,  the  arc;nments  1 — 9,  and  the  times  with  which 
Table  XVII  is  to  be  entered,  for  the  intermediate  dates,  by  adding  the  nearly 
constant  differences  dcdnced  from  the  numbers  at  the  bottom  of  Table  II.  The 
seconds  of  <j  arc  first  reduced  to  fractions  of  a  minute,  with  which  to  enter  Table 
VII.  In  makinfT  the  subsequent  computation  we  have  used  none  of  the  devices 
previously  described  except  in  the  case  of  the  small  longitude  terms,  as  follows : 


THE   OUBIT  OF  URANUS. 


201 


0 

o 

0 

0 

0 

0 

o 

9 

330 

332 

334 

330 

338 

340 

342 

8? 

270 

270 

2H2 

288 

204 

300 

son 

iy 

240 

248 

250 

204 

272 

280 

288 

1) 

n 

>> 

/' 

II 

II 

/' 

/,, 

.(..3) 

—  1.14 

—  1.13 

—  1.13 

—  l.ir, 

—  1.23 

—  1.33 

—  1.48 

(•' 

.c'.:t) 

—  2.4fi 

—  a-.-io 

—  2.  CO 

—  2.7(1 

—  2.88 

—  3.01 

—  3.10 

(l 

.^.4) 

-f  o.r.o 

-f  O.fiO 

+  0.,'')0 

+  0.55 

+  0.50 

+  0.43 

+  0.30 

(l 

.f.4) 

—  0.10 

—  0.20 

—  0.30 

—  0.40 

—  0.47 

—  0.57 

—  0.04 

(l 

.X.3)  Bill  3;; 

+  1.14 

+  1.12 

+  1.11 

+  1.11 

+  1.12 

+  1.15 

+  1.20 

(t 

.(•.;i)  COS  It;/ 

.00 

—  0.20 

—  o.,^5 

—  0.85 

—  1.17 

—  1.50 

—  1.80 

(l 

.0.4)  sill  4;/ 

—  .,11 

_    .M 

—   ..17 

—    .55 

—    ..50 

—    .42 

—    .34 

(l 

.r.4)  ecis  4f7 

4-  .0,'j 

+    .07 

+    .07 

+    .04 

—    .02 

—    .10 

_    .19 

Siiiu 

+  O.fiS 

+  0.37 

+  0.00 

—  0.25 

—  0.57 

—  0.87 

—  1.19 

It  will  bo  s(>en  tliat  wo  hiivo  licro  romputod  twicu  as  many  numbers  as  arc 
nocos.sary  to  intcrpoliito  with  all  attainable  aixuiacy. 

Tiio  rest  of  tiio  computatiim  is  fully  jjivcn  on  the  four  followin«,'  paj^os.  First 
wo  have  the  values  of  ij  and  the  nine  arf,'umonts  for  the  intermediate  dates,  filled 
in  by  suecessivo  additicm  of  the  nearly  constant  diflerence.  The  ar<,'uinents  thus 
obtained  for  the  last  date  may  be  compared  with  those  just  cimiputed  on  the  pre- 
ceding page. 

The  numerals  in  tin;  first  columns  of  the  sceticms  of  computation  following 
indicate  the  argum(-nts  with  which  tables  arc  entered  to  obtain  tlie  separate  values 
of  the  quantities  (r.c.O),  (r.«.l),  (r.c.l),  etc.  Tlio  negative  terms  in  Table  XVII 
being  taken  from  the  sum  of  all  the  periodic  terms  from  Tables  VIII  to  XVI  with 
argument  1  to  9,  we  have  the  final  values  of  {r.c.O),  {v.s.l),  etc. 

The  final  computation  of  the  products  («.«.(')  sin  i;/,  etc.,  and  the  addition  of  the 
separate  terms  which  make  up  the  three  co-ordinates,  are  shown  on  page  1205.  The 
expressions  c.O,  .s.l,  etc.,  are  employed  for  brevity,  instead  of  (r.c.O),  {v.s.l)  sin  </,  etc. 

The  lon"itude  finally  given  by  the  tables  is  referred  to  the  mean  equinox,  and 
must  therefore  be  corrected  for  nutation  before  bouig  used  to  compute  the  geocentric 
place. 


June,  1873 


IPs 


909 


TI!  !•:   OIIBIT   OF   URANUS. 


D.l«,     1 

l»7.'>,  l>H«.  16  I87'l,  Apr    i:i 

Auk    M 

Di'c.  » 

Xm,  Apr.  H 

Aiii.'    H 

!>..•,  4 

1>7K.  Apr  3 

l»75.tf5D. 
0           / 

l»7(i.'>3. 

lb7ii  lili. 

lb7i).U4(l. 

11)77. ilijtf. 
0            / 

In77,r)li7, 

0               f 

lh77. !•-.'>. 
0            / 

0               / 

0          / 

o           / 

0           / 

Hal    23. 081! 

332   47.554 

331    12.021 

335    30.480 

337      0,!»57 

338    25,425 

339    49.803341    14.3(ll| 

ArK.   1 

U-.(t!l7 

ini.ac.s 

175.(13!) 

I80.!(l(t 

204.182 

218,453 

232.724 

240.0!t5 

2 

32ll.H,'ii'> 

32.'). (Mil 

aj',i.34(l 

333.(102 

3as.(i;is 

342.383 

34(1.720' 

351,(174 

a 

iT.i.-ai) 

1H0.«.S!( 

1^2.038 

l83.  1M8 

ls.»,y;i8 

1.S5.488 

18(1.038 

187.788 

4 

\:>:i 

150. 

150. 

1(13. 

ico. 

109. 

172. 

175. 

5 

lea. 

•   •    •    .    • 

•  .  .   .   • 

175. 

ts 

4S1). 

4!H. 

500. 

505. 

511, 

5io. 

521. 

527, 

1 

i-.'a. 

12!(. 

135. 

141. 

117, 

153. 

ICO, 

100, 

8 

4U1.C 

403.4 

405.2 

407.0 

408,7 

410.5 

412.3 

414.1 

•1 

(D.O.O)  1 

6(1. 

(13. 

(10. 
10(1.2!) 

75. 
103.08 

82. 

88. 
95.25 

95. 
89,02 

101. 

83.21 

los.ol 

107.74 

0!l,il8 

2 

ir,..so 

14.71 

13.(1(1 

12.(15 

11,08 

1((,70 

0,00 

0,00 

S 

72.4a 

73.(10 

74.77 

75.04 

77.10 

78.20 

70.42 

80.50 

0 

.,■.;> 

.58 

.58 

.57 

.50 

.50 

.55 

,54 

l> 

.(12 

.02 

.02 

.03 

.03 

.03 

.03 

.03 

7 

.07 

.07 

.0(1 

.00 

.00 

,00 

.05 

.05 

8 

l.CS 

1.(15 

i.(ia 

1.00 

1.58 

1.55 

1.53 

1.50 

(.'.<-.0) 

.011 

I'jH.di) 

.011 
li)8.4(i 

.00 
107.10 

.00 
lft4.02~ 

.08 

.08 

.08 

.07 
175.05 

101.07 

180.55 

181.18 

(r.«.l)l 

fi.84 

7.(18 

8.40 

8.00 

9.44 

9.74 

9.01 

0.08 

:! 

132.20 

12.').48 

118  82 

112.21 

105.07 

99.2» 

92.89 

80.09 

SVC.     i 

.IS 

.  20 

•)•) 

.23 

.  25 

.27 

.29 

.31 

:i 

l.(l!l 

1.07 

1.08 

1.12 

].10 

1.20 

1.41 

1.50 

4 

.13 

.12 

.11 

.10 

.10 

.00 

.09 

.OX 

f) 

.!'.» 

.10 

.10 

.10 

.lit 

.20 

,20 

.20 

ii 

.04 

.04 

.04 

.05 

.05 

.00 

.00 

.07 

7 

.14 

.14 

.13 

.13 

.12 

.12 

.11 

.10 

H 

1.28 

1.28 

1.27 

1.2(1 

1,20 

1.25 

1,24 

1,23 

y 

z 

.OS 
142.17 

.08 
13r>.28 

.08 

130.34 

.08 

,07 
118.34 

.07 
112.33 

.07 
100.27 

.07 

124.30 

100.20 

Tab.  XVII 

(r.,..l) 

(u.e.l)l 

— ir)4.21 

—153.84 

_lf,;j,47_ 
—  23.13 

0.38 

—  153.10 

—  28.74 

—152.73 

—152.30 

—151  99 
—  45.72 

— 151.(12 
—  51.33 

—  12.04 

—  17.5(1 
fi.03 

—  34.30 

—  40.03 

5.fi4 

0.07 

0.85 

0.01 

0.S2 

0.59 

•2 

5.01 

fi  00 

7-31 

8.00 

10.80 

13.00 

15.49 

18.25 

soc.  •! 

.CI) 

.71 

.72 

.74 

,75 

.77 

.79 

.so 

••i 

23.20 

24.00 

24.05 

25.82 

20,08 

27.53 

28.37 

20. 1 0 

4 

.47 

.48 

.40 

.50 

.50 

.51 

.frtJ 

.52 

f) 

.37 

.37 

.37 

.:;7 

.37 

.37 

.37 

.37 

(i 

.1(1 

.17 

.17 

.18 

.18 

.18 

.18 

.18 

7 

.1.-. 

.14 

.13 

.13 

.12 

.12 

.11 

.10 

8 

0.  (iC. 

•      0.(14 

0.(13 

0.01 

0.00 

0.50 

0,57 

0.50 

.07 
3(!.42 

.07 
38.70 

.0(1 
41.21 

.00 

.((5 

.04 

.04 

.03 
50,50 

43.08 

40.00 

50,02 

53.20 

TuIlXVII 

— 20.').  hi; 

— 20(1.08 

—20(1.20 

—200.51 

—200.73 

—200.04 

—207,15 

—207,37 

(f.r.l) 

— 1(',!).44 

—  1(17.38 

—105.08 

—102.53 

—  150.83 
.20 

—  150.92 

—  153.89 

—  150.78 

(W.S.2)  1 

.or. 

.10 

.15 

.21 

.31 

.34 

.37 

2 

54.12 

40.37 

44.78 

40.37 

30,15 

32.11 

28.30 

24.70 

sec.  2 

.3!) 

.42 

.44 

.47 

.40 

.51 

.53 

.50 

3 
2 

2.05 

2.19 

2.31 

2.45 

2.00 

2.70 

2.91 
32,08 

3.07 
28.70 

5r..fi2 

52.08 

47.08 

43.50 

30.50 

35.00 

Tuh.XVII 

—1.34.34 

—134.32 

—134.31 

—  80.03 

—134.20 

—134.27 
—  04.77 

—  134.20 

—  08.57 

—  134.24 
— 102.i0 

—134.22 
—  105.52 

|(.-..s.2) 

—  77.72 

—  82,24 

_  00.79 

T  H  E  0  II IJ  IT  OF   U  11 A  N  U  S 


W3 


l)M», 


iHT.'i,  D.'O.    ID 
Ih7li,l»,'j6. 


Ih7i).  Apr.   l:(,       Aux.  11      I       Of,'.  » 
Itilii.M.i.    \     lb7il  UU.    I     lf<U7.»4ll. 


9 

(u.c.2)  1 

2 

IOC.   2 

3 

S 
Tall.  XVI 1 

(«,o.2) 

(u.«.3)  2 
3 

Tub.  XV 11 
(o.tr.3) 


331    23.0S(> 

.31 

2">  34 

.711 

0.23 

32.(14 
-131). .'>7 

-IU3.03 


(u.0.3)  2 
3 


Tab.  XVII 

(o.f.3) 


(u.hA)  2 
3 

(v.sA) 


0.02 

0.H7 

7.70 

— «.02 

—1.13 

8.19 

1.34 

4.. '^3 
-7.0(1 


O  >  O  /  0 

132   47.. '154  334    12.02133:)    3(1. 4S0 


.30 

2H.1)3 

."(1 

(1.40 

3(1.30 
—  13(i.fi0 

—100.20 

o.so 

0,01 


7. HO 
— H.02 

—  1.12 

3.12 
1.34 

4.4(1 
—".(((1 

— 2.r)3 

1.4S 

— OSS 

-|-o.(;(» 


—2.00 

1.4S 

— O.SH 

-f  0.(10 


(u.c.4)2 
3 

(■U.C.4) 


(p.<.'.0)  1 
2 

3 
Tab.XVll 

(ff^) 

(p.».l)  1 
.J 

3 

S 
Tab.XVll 

(p.S.l) 


0.S3 
—  1.00 

—0.17^ 


1230 
OS 
11 

0(18 


(p.c.l)  1 
2 
3 


Tab.XVll 

(p.c.l) 


2307 

248 

2S.J-) 

173 

32,')(1 
—  10S4 

+  1272 


123 

12(1(1 
CO 

14,'>5 
-1077 


0.77 
—1.02 

-0.2') 


10(13 

00 

10 

0(18 


2140 


240 
2S22 

KIH 

3230 
-10S7 

1243 


.20 

32.71 

,77 

(1,5(1 

40,33 

—  13(1,(10 

—  0(1.27 


112 

1202 
66 


13H0 
-lOSl 


f..H4 
_    0.04_ 

7.78 
—8.01 

—  1.13 


3.05 
l.>14_ 

4,30 
—7.0(1 

—2.(17 


.20 
36.73 

.78 
6.70 

44.50 
—13(1.62 

—  02.12 

6,78 
0,08 

'^^    7.76 

—8.01 
~^1.15" 


2.08 
1.34 

4.32 
—7,06 

—2.74 


1.47 

— 0.H8 

-+-0.50 


0.72 
—1.03 

—0.31 


800 

lol 

0 

9(18 

"l077 


522  —  601 


230 

2S07 
1(15 

3202 
-1080 

1213 


07 
1138 

67 


1.302 
-1086 

-  684 


1.45 

-0.80 

-f  0.5(1 

0.66 
—1.04 

-0.38 


741 
104 

0 
967 

~i82^r 


221 

2780 
161 


lh77,  Apr.  H  I 
l»77.l!tl!».     I 


AiiK.  II 
lh77  f.l>7. 


IIhii.  4 
Ml  MM. 


o  /         I    o  >  o  / 

U37      0.057  338    25.425  339    49.893 

.20 

40.03 

.70 

6.85^ 

48,86 
—136.64 

—  87.78 


6,70 
1,01 

7.71 

— H.Ol 

—  1.20 


.30 

45.35 

.79 

(1.99 

63.43 
—136.06 

30 

40  03 

.HO 

7.10 

68.13 
—136.67 

—  83.28 

—  78.64 

6.61 
1.04 

6.51 

1.08 

2,00 
1,33 

4,23 
-7.0(1 

-2.83 


1,43 

— o.si) 

4(1.54 


0.61 
—1.05 

—0.44 


504 
108 

s 
967 

1677 

211 

2760 
158 


3171 

-1001 

1180 


3138 
-1 003 

1145 


83 

1074 

67 

1224 
—1000 

—  766 


69 

1000 

68 

1146 
—1094 

—  848 


7.65 
—8.91 


2.82 
_l,33 

4,15 

—7.0(1 

—2.01 


7.50 
—8.01 

"—1.32 


2.75 
1.31 

4.06 
-7.0(1 


—3.00 


1.40 
—0.01 


+0.40 

0.57 

—  1,0(1 

—0.40 

460 
113 

8 
967 

1548 

203 

2748 
155 

3106 

—  1006 

~^  1110 

55 

047 

70 

1072 
-1098 

—  926 


3.36 
—0.03 

+  0.43 


0.51 
—  1.07 

—0.56 


314 

118 

8 

06(1 


I»<78,  Apr   3 
IH-M  2!i4. 


341    14.361 

.31 

54.60 

.NO 

7,20 

63.00 

—  136.69 

—  73.69 


6.37 
1,11 

7.48 
—8.01 


1436 


104 

2723 
151 

3068 
-1008 

1070 


-1.43 


2.67 
1.29 

3.06 
-7.06 

-3.10 


1.33 
_  .04 

+0.30 


0.47 
—  1.08 

—0.61 


246 

126 

9 

966 


1347 


187 

2606 

148 

3031 
2000 

1031 


41 

886 
72 


31 

827 
74 


909 
-2002 

-1003 


932 
2006 

—1074 


ilftl 


i-'<i 


204 


THE   ORBIT  OF  URANUS. 


T)ate,    1 

1875,  Dot.  16 
1 875.1.(65. 

187fi,  Apr.  13 
1870.283. 

o          / 

332   47.654 

181 
25 

Aug.  11 
l87ti.Uia. 

o           / 

334    12.021 

189 
25 

lief.  0 
18(i7.»40. 

o          / 

335   .36.489 

196 
24 

4l>77.  Apr.  8 
lt>77.2ii». 

o            ' 

337      0.!)57 

204 
24 

228 
—460 

—232 

Ann.  6 
1877.597. 

Dfo.  4 
1877.925. 

1878,  Apr.  3 
1878.264. 

9 

(p.8.2)  2 
3 

S 
Tab.  XVI I 

(p.8.2) 

o          / 

331    23.0S(i 

174 
2fi 

o         r 

.338   25.425 

213 
24 

o           / 

339   49.!- 93 

223 
23 

o           / 

.341    14.361 

232 
23 

200 
—459 

20C 
—459 

2i4 

—459 

220 

—459 

—239 

237 
—460 

~— 223~ 

246 
—460 

255 

—460 

—259 

—253 

—245 

576 
34 

—214 

—205 

(p.c.2)  2 
3 

S 
Tab.  XVI I 

(p.c.2) 

559 
33 

5fiS 
33 

584 
35 

6!)2 
36 

600 
37 

607 
39 

614 
40 

592 
-404 

+  128 

601 

—464 

610 

—464 

619 

—465 

628 
—465 

n;3 

637 
—465 

646 

—465 

654 
—466 

137 

146 

154 

39 
—101 

172 

181 

188 

(p.s.3)  2 
Tab.XVII 

(p.8.3) 

(-^.^.3)  2 
Tab.XVII 

(p.c.3) 

28 
—101 

31 

—101 

34 
—101 

43 

—  101 

47 

—  101 

52 
—101 

57 
—101 

—  73 

—  70 

—  67 

—  62 

—  58 

—  54 

—  49 

175 
—102 

73 

—  44 

155 
— lOJ 

+  53 

0.12 
0.11 
0.07 

0.30 

159 
—102 

57 

162 
—102 

166 
—102 

169 
—102 

67 

172 

—102 

177 
—102 

75 

60 

64 

70 

(6.C.0)  2 
.'t 
Tab.XXIIl 

(6.C.0) 

0.12 
0.11 
0.07 

0.13 
0.11 
0.07 

0.13 
0.11 
0.07 

0.31 

0.14 
0.11 
0.07 

0.32 

0.14 
0.11 
0.07 

0.32 

0.15 
0.11 
0.07 

0.15 
0.11 
0.07 

0.30 

0.31 

0.33 

0.33 

(6.S.1)  1 
•2 

3 

Tab.XXlIl 

(6.S.1) 

0.30 
0.12 
l.ll 

1.53 
—5.23 

—3.70 

1.21 
O.'.X! 
1. 01 

0.22 
0.09 
1.11 

1.42 

—5.24 

—3.82 

0.16 
0.06 
1.12 

0.12 
0.05 
1.12 

0.08 
0.04 
1.13 

1.25 
—5.24 

—3.99 

0.06 
0.04 
1.13 

1.23 

—5.25 

4.02 

0.05 
0.05 
1.14 

1.24 

—5.25 

0.06 
0.06 
1.14 

1.26 
—5.25 

— 3.99 

1.34 

—5.24 

3.90' 

1.06 
1.17 
1.00 

3.23 
—4.43 

—1.20 

1.2!) 
—5.24 

—3.95 

—4.01 

(6.^.1)1 
2 
3 

S 
Tab  XXIII 

(6.P.1) 

1.14 

1.06 
1.00 

0.!)8 
1.28 
0.99 

3.25 
—4.43 

—  1.18 

0.04 
0.15 

—0.48 

—0.29 

0.34 

0.10 

—0.38 

+0.06 

0.!10 
1.39 
0.99 

3.28 
—4.42 

— 114 

0.80 
1.51 

0.99 

3.. 'id 
4.41 

—1.11 

0.70 
1.63 
0,98 

3.31 
—4.4  1 

—1.10 

O.fil 
1.75 

0.98 

3.1:i 
—4.45 

-1.27 

3.20 
—4.44 

—1.24 

3.34 
—4.40 

—  1.06 

(6.8.2)  2 
3 
Tab.XXIIl 

(6.8.2) 

0  02 

O.K! 

— 0.4S 

-0.30 

0.31 

0.10 

— 0.3>i 

4-0.03 

0.02 

0.16 

—0.48 

—0.30 

0.03 

0.16 

-0.48 

—0.29 

0.04 

0.15 

—0.48 

—0.29 

0.05 

0.15 

—0.48 

—0.28 

0.06 

0.15 

—((.48 

~— 0.27 

0  30 
0.11 

—0.38 

+0.09 

0.07 

0.15 

—0.48 

—0.26 

(6.C.2)  2 
3 
Tab.XXIIl 

(6.„.2) 

0.32 
0.10 

—0.38 

0.33 
0.10 

—0.38 

f0.05 

0.34 
0.11 

—0.38 

0.35 
0.11 

—0.38 

0.36 
0.11 

—0.38 

40.09 

+0.04 

+0.07 

+  0.08 

THE   ORBIT   OF   URANUS, 


205 


Date,    1 

187.'),  D™.  15 

;  Ih".   ,  April  13 

1        AnK-  11 

Dtto.  9 

1877,  April  8 

A  UK-  •! 

Deo.  4 

1878,  April  3 

IbTS.aOO. 

1      lbTti.2»3. 

1       lH7ti.(il2. 

lb7li.94U. 

lS77.2lii). 

1877.697. 

1877.925. 

1878.2S4. 

0          r 

0            / 

1      °           ' 

0           / 

o           / 

0           / 

o          / 

O              1 

9 

331    23.080 

333   47.554 

334    12.021 

335   30.489 

337     0.957 

338    25.425 

339    49.893 

341    14.301 

Iog(u.8.1) 

— i.osor. 

—1.2445 

—1.3042 

—1.4585 

—1.5305 

—1.0024 

-1.6001 

—1.7104 

8111  g 

_9.(;ho3 

— 9.(;(;oi 

—9.0380 

—9.0159 

—9.5910 

—9.50.55 

—9.5375 

—9.5073 

l0g(p.8.1) 

+3.1045 

3.0944 

3.0838 

3.0719 

3.0588 

3.0453 

3.0294 

3.0132 

log(u.e.l) 

—2.2290 

—2.2237 

—2.2177 

.^2.210 

—2.20.30 

—2.1957 

—2.1872 

—2.1784 

cos  7 

+  9.9434 

9.9491 

9.954 

9.9594 

9.9041 

9.9085 

9.9725 

9.9703 

log(p.f.l) 

—2.718 

—2.779 

—2.835 

—2.884 

—2.928 

—2.907 

—3.0013 

—8.0310 

l0)r(«.,S.2) 

—  1.890.') 

—1.9151 

— 1. 9370 

—1.9580 

-1.9700 

—1.9937 

—2.0093 

—2.0233 

sill  iff 

—9.9247 

— 9...102 

—9.8941 

—9.8703 

— 9.85(;0 

—9.8350 

—9.8111 

—9.7840 

log(p.s.2) 

—2.413 

—2.403 

—2.389 

—2.378 

—2.305 

—2.348 

—2.330 

—2.312 

log(u.,..2) 

— 2.01(;7 

—2.0009 

—1.98.35 

—1.9044 

—1.9434 

—1.9202 

—1.8951 

—1.8074 

CDS  2(/ 

+9.7335 

9.7049 

y.7932 

9.8189 

9.8421 

9.8030 

9.8821 

9.8994 

K>fr(p.<-.2) 

+2.107 

2.137 

2.104 

2.187 

2.212 

2.230 

2.258 

2.274 

o      '        n 

0        ■          '/ 

O         f 

0      /       '.• 

0        t          ft 

0       /         n 

O        /          II 

0       1        II 

A!7 

1  24  28.0(15 

1  24  2S.O00 

1  24  ':>..000 

1  24  28.007 

1  24  28.008 

1  24  28.008 

1  24  28.009 

a 

331  23    fj.lO 

332  47  33.225 

334  12     1.291 

3,35  30  29.357 

337    0  57.424  338  25  25.492 

339  49  53.500 

341  14  21.63 

u 

95  11  41.00 

95  11  61.40 

95  J2     1.81 

95  12  12.21 

95  12  22.01 

95  12  33.01 

95  12  43.42 

95  12  53.82 

E 

—2  42  49.15 

—2  35  33.18 

—2  28  10.i;3 

—2  20  41.70 

—2  13    0.93 

—2    5  20.43 

— 1  57  40.55 

— 1  49  49.00 

c.O 

3  18.i;9 

3  18.40 

3  17.10 

3  14.02 

3  11.07 

3    6.55 

3    1.18 

2  55.05 

8.1 

6.77 

8.03 

lO.Oi 

11.87 

13.43 

14.72 

15.70 

10.51 

e.l 

—2  28.72 

—2  28.80 

—2  28.02 

—2  28.00 

—2  27.13 

—2  25.93 

—2  24.43 

—2  22.80 

«.2 

1     5.34 

1     0.88 

1    7.87 

1     8.28 

1     8.12 

1    7.40 

1     0.13 

1    4.20 

r.2 

—    50.27 

—    68.31 

—    59.. '^O 

—1    0.71 

—1     1.03 

—1     0.70 

—    69.87 

—    58.45 

(3  +  4) 

0.40 

0.25 

0.03 

—      0.19 

—      0.41 

—      0.03 

—      0.83 

—      1.00 

11 

03  .'■)3    2.28 

05  24  57.89 

00  50  59.12 

08  29    5.08 

70    1  17.15 

71  33  33.48 

73    5  54.37 

74  38  19.30 

e 

73  22    0.15 

73  22    0.20 

73  22  12.37 

73  22  IS. 48 

73  22  24. CO 

73  22  30.71 

73  22  30.. s2 

73  22  42.93 

R 

2.5,-i 

2.91 
138  47     7.00 

1.2(!44735 

3.25 

3.01 

3.98 

4.37 

4.79 

6.22 
143    1    7.61 

Ldiigitudc 

l.n  15    5.01 

140  19  14.74 

141  51  27.77 

143  23  45.73 

144  50    8.50 

140  28  35,98 

l"f?  '•o 

1.2i;4-392 

1.2042190 

1.2(i39778 

1.2037480 

1.2035319 

1.2033280 

1.2031370 

c.O 

2307 

2140 

1977 

1821 

1077 

1548 

1436 

1347 

s.l 

—009 

—508 

—528 

—487 

—447 

—408 

—309 

—  332 

e.l 

—458 

—535 

—015 

—097 

—780 

—801 

—941 

—1017 

S.2 

+  218 

200 

192 

179 

107 

152 

138 

12: 

C.2 

+   09 

80 

91 

101 

113 

120 

138 

149 

8.3 

+  73 

09 

05 

CO 

64 

49 

43 

37 

C.3 

+     •* 
1.2048990 

8 

13 

19 

U 

29 

80 

41 

log  r 

1.2040135 

1.2043391 

1.2G40774 

1.2038293 

1.2035954 

1.2033701 

1.2031720 

1                H 

/       II 

'       // 

>       It 

'       // 

3« 

+  41    30.04 

42    8.4S 

42    .'IS. 54 

43    0.78 

43    33.14 

43    57.02 

44    20.23 

44    41.19 

c.O 

4().:!0 

0.30 

0..I1 

0.31 

0.32 

0.32 

0.33 

0.33 

.'.■l 

+  1.77 

1.75 

1.70 

1.03 

1.57 

1.48 

1.38 

1.29 

e.l 

—1.11 

—1.1(1 

—1.08 

—1.07 

—  1.05 

—1.03 

—1.03 

—  1.00 

8.2 

+0.25 

0.21 

0.23 

0.22 

0.21 

0.19 

0.17 

0.10 

0.2 
Latitudo 

+  0.02 

0.03 
42    9.70 

0.03 
42   39.73 

0.04 
43    7.91 

0.05 

0.06 

0.07 

0.07 
44   42.04 

+  41    37.87 

43   34.24 

43   68.04 

44    21.15 

li: 


206 


THE   ORBIT   OF   URANUS. 


TABIiE 

I. CoUEKCTIONS  OF  AKOUMENTS  TOR  PAST  AND  FUTURE  t'ENTURl 

E8. 

Arg.  1 

Century. 

9 

u 

u 

6 

0' 

O 

/ 

tt 

0 

t 

// 

// 

o 

1        It 

tt 

OJ 

207 

15 

59.32 

343 

52 

17.36 

+  108.00 

351 

C  26.89 

—148.32 

408.924 

100 

275 

45 

34.18 

344 

40 

54.12 

102.00 

351 

34  55.39 

—140,08 

552.952 

200 

344 

15 

9.04 

345 

41 

24.88 

96.00 

352 

3  32.13 

—131.84 

96.980 

300 

52 

44 

43.90 

346 

35 

49.64 

90.00 

352 

32  17.11 

—123.00 

241.008 

400 

121 

14 

18.76 

347 

30 

8.40 

84.00 

353 

1   10.33 

—115.36 

385.036 

500 

180 

43 

53.02 

348 

24 

21.10 

-1-   78.00 

353 

30  11.79 

—107.12 

529,004 

COO 

258 

13 

28.48 

349 

18 

27.92 

72.00 

£j3 

59  21.49 

—  98.88 

73.092 

700 

326 

43 

3.34 

350 

12 

28.08 

60.00 

354 

28  39.43 

—  90.64 

217.120 

800 

35 

12 

38.20 

351 

6 

23.44 

00.00 

354 

58     5.61 

—  82.40 

301.148 

900 

103 

42 

13.06 

352 

0 

12.20 

54.00 

355 

27  40.03 

—  74.16 

505.170 

1000 

172 

11 

47.92 

352 

53 

54.96 

+  48.00 

355 

57  22.69 

—  05.92 

49.204 

1100 

240 

41 

22.78 

353 

47 

31.72 

42.00 

356 

27   13,59 

—  57.08 

193.232 

1200 

309 

10  57.64 

354 

41 

2.48 

30.00 

356 

57   12.73 

—  49.44 

337.200 

1300 

17 

40 

32.50 

355 

34 

27.24 

30.00 

357 

27  20.11 

—  41.20 

481.288 

1400 

80 

10 

7.36 

356 

27 

40.00 

.      24.00 

357 

57  35.73 

—  32.96 

25.310 

1500,1 

154 

39 

42,22 

357 

20 

58.70 

+   18.00 

358 

27  59.59 

—  24.72 

169.344 

15000 

154 

32 

39.89 

357 

20 

57.89 

18,00 

358 

27  59.08 

—  24.72 

168.1.55 

1(100 

22.1 

2 

14.75 

358 

14 

4.05 

12.00 

358 

58  31.18 

—  16.48 

312,1.H3 

1700 

291 

31 

7.37 

359 

7 

5.33 

0.00 

359 

29  11.47 

—     8.24 

450,092 

1800 

0 

0 

0.00 

0 

0 

0.00 

0.00 

0 

0     0.00 

0.00 

0.000 

1000 

08 

28 

52.63 

0 

52 

4S.67 

—     6.00 

0 

30  56.7V 

+     8.24 

143,908 

2000 

130 

58 

27.49 

1 

45 

31.43 

—  12,00 

1 

2     1.83 

16.48 

287,936 

2100 

205 

27 

20.11 

2 

38 

8.11 

—  18.00 

1 

33   15,08 

24.72 

431.845 

2200 

273 

56 

12,74 

3 

30 

38.78 

—  21.00 

2 

4  30.57 

+  .32.96 

575.755 

TABLK 

11— A 

HaUMKNTS  FOR  THE  BEQINNINQ  OK  EACH  FoURTH 

Year  17 

52—1948. 

Year. 

9 

U 

0 

Arg.  1 

o 

1 

ti 

o 

t 

ft 

O 

t 

tt 

1    ,        1752 

160 

15 

8,10 

94 

6 

10,48 

7r 

43 

42.30 

102.101 

1756 

177 

23 

31.10 

94 

8 

17.46 

1  *d 

44 

56.25 

335.8(12 

1760 

194 

31 

54.09 

94 

10 

24.43 

72 

40 

10.22 

509,023 

1704 

211 

40 

17.09 

94 

12 

31,39 

72 

47 

24.21 

83,384 

1708 

228 

48 

40.08 

94 

14 

38.34 

7i 

"H 

38,20 

257.115 

1772 

245 

57 

3.08 

94 

10 

45.28 

72 

49 

.52.21 

430.907 

1776 
■           1780 

263 

5 

20.07 

94 

18 

52.22 

72 

51 

0.23 

4.068 

280 

13 

49.00 

94 

20 

59,14 

72 

52 

20,27 

178.429 

1784 

297 

22 

12.00 

94 

23 

6.05 

72 

53 

34.32 

352,190 

1788 

314 

30 

35.05 

94 

25 

12.96 

72 

54 

48,38 

525,951 

1792 

331 

38 

58.05 

94 

27 

19,85 

72 

56 

2,40 

99,712 

1796 

348 

47 

21,04 

94 

29 

26,74 

72 

57 

16  54 

273.473 

1800 

6 

55 

1,.>S0 

94 

31 

33,53 

72 

5H 

3(1,59 

447,115 

■ 


TlIK    ORBIT    OF    URANUS. 


207 


TA15LE  I 

. — Continued. 

Century. 

2 

3 

4 

5 

6 

7 

8 

9 

OJ 

191.528 

299.485 

280 

512 

102 

61 

470.0 

15 

100 

314.245 

49.520 

04 

250 

503 

124 

410.6 

214 

200 

430.902 

399.555 

449 

589 

424 

188 

351.2 

413 

300 

559.079 

149.590 

233 

327 

285 

251 

291.7 

13 

400 

82.396 

499.025 

18 

05 

146 

314 

232.3 

212 

500 

205.113 

249.600 

402 

403 

1 

378 

172.9 

411 

fiOO 

327.830 

599.095 

187 

142 

468 

441 

113.5 

10 

700 

450.547 

349.730 

571 

480 

329 

604 

54.1 

209 

800 

573.204 

99.705 

356 

218 

100 

507 

594.6 

409 

900 

95.981 

449.800 

140 

550 

51 

31 

536.2 

8 

1000 

218.098 

199.835 

624 

295 

512 

94 

475.8 

207 

1100 

341.415 

649.870 

309 

33 

373 

157 

410.4 

400 

1200 

404. 1?2 

299.905 

93 

.•^il 

234 

221 

357.0 

5 

13110 

680.849 

49.940 

478 

109 

95 

284 

297.6 

205 

1400 

109.506 

399.975 

202 

448 

660 

347 

238.1 

404 

1500  J 

232.283 

150.010 

47 

186 

417 

411 

178.6 

3 

1500CJ 

231.921 

149.914 

47 

186 

417 

410 

178.3 

2 

1000 

354.038 

499.949 

431 

524 

278 

473 

118.8 

201 

1700 

477.319 

249.975 

216 

202 

139 

537 

59.4 

401 

1800 

0.000 

0.000 

0 

0 

0 

0 

0 

1 

0 

11100 

122.081 

350.025 

385 

338 

461 

63 

540.6 

199 

2000 

245.398 

j  100.000 

109 

76 

322 

127 

,  481.2 

398 

2100 

308.079 

460.080 

554 

415 

183 

IflO 

421.8 

697 

2200 

490.700 

200.111 

338 

153 

44 

253 

'  362.3 

196 

TABLE  U.  — Continued. 

Year. 

2 

3 

4 

5 

6 

229 

7 

8 

9 

1762 

481.104 

346.855 

133 

348 

213 

331.6 

678 

1766 

534.013 

369.856 

172 

362 

296 

287 

353.2 

58 

1760 

586.921 

373.858 

212 

375 

362 

302 

374.8 

138 

1704 

39.830 

387.869 

251 

389 

429 

436 

396.4 

218 

1768 

92.739 

401.860 

200 

402 

495 

511 

418.1 

298 

1772 

145.047 

415.862 

330 

416 

562 

585 

439.7 

377 

1776 

198.666 

429.803 

369 

429 

28 

60 

461.3 

467 

1780 

251.465 

443.865 

408 

443 

95 

134 

482.9 

537 

1784 

304.373 

467.866 

448 

467 

161 

209 

504.6 

17 

1788 

357.282 

471.868 

487 

470 

227 

283 

526.2 

97 

1792 

410.191 

485.860 

527 

484 

204 

358 

547. S 

177 

1790 

403.100 

499.870 

566 

497 

360 

433 

569.4 

257 

1800 

515.972 

513. S62 

6 

611 

427 

607 

691.1 

337 

208 


THE   ORBIT   OF   URANUS. 


TAHLE 

II. — Continued. 

Year. 

V 

(J 

0 

Arg.  1. 

0 

/ 

/' 

0 

t          i> 

0 

/ 

tt 

1800 

5 

55 

1.8m 

94 

31  33.53 

72 

58 

30.59 

447.115 

1801 

23 

3 

24.80 

94 

33  40.39 

72 

59 

44.71 

20.876 

1808 

40 

11 

47.79 

94 

35  47.25 

73 

0 

58.84 

194.638 

1812 

57 

20 

10.79 

94 

37  54.09 

73 

2 

12.98 

368.399 

18IG 

74 

28 

33.78 

94 

40   0.93 

73 

3 

27.13 

542.100 

1820 

91 

36 

56.77 

94 

42   7.76 

73 

4 

41.30 

115.921 

1824 

108 

45 

19.77 

94 

44  14.58 

73 

5 

65.48 

289.683 

1828 

125 

53 

42.76 

94 

46  21.38 

73 

7 

9.67 

463.444 

1833 

143 

2 

5.76 

94 

48  28.18 

73 

8 

23.88 

37.205 

1830 

160 

10 

28.75 

94 

50  34.97 

73 

9 

38.10 

210.966 

1840 

177 

18 

51.75 

94 

52  41.75 

73 

10 

52.33 

384.727 

1844 

194 

27 

14.74 

94 

54  48.52 

73 

12 

6.58 

558.488 

1848 

211 

35 

37.74 

94 

56  55.28 

73 

13 

20.84 

132.249 

1852 

228 

il 

0.73 

94 

59   2.03 

73 

14 

35.11 

306.010 

185G 

245 

6a 

23.7b 

95 

1   8.77 

73 

15 

49.40 

479.771 

1800 

203 

0 

46.72 

95 

3  15.50 

73 

17 

3.70 

53.532 

1804 

280 

9 

9.71 

95 

5  22.22 

73 

18 

18.01 

227.293 

1868 

297 

17 

32.71 

95 

7  28.94 

73 

19 

32.34 

401.054 

1872 

314 

25 

55.70 

95 

9  35.64 

73 

20 

46.67 

674.815 

1876 

331 

34 

18.70 

95 

11  42.33 

73 

22 

1.03 

148.576 

1880 

348 

42 

41.69 

95 

13  49.01 

73 

23 

15.39 

322.337 

1884 

5 

51 

4.69 

95 

15  55.69 

73 

24 

29.77 

496.098 

1888 

22 

59 

27.68 

95 

18   2.35 

73 

25 

44.16 

69.860 

1892 

40 

7 

50.68 

95 

20   9.01 

73 

26 

58.57 

243.621 

1896 

57 

16 

13.67 

95 

22  15.05 

73 

28 

12.98 

417.382 

1900 

74 

23 

54.43 

95 

24  22.20 

73 

29 

27.36 

691.024 

1904 

91 

32 

17.43 

95 

26  28.83 

73 

30 

41.81 

164.785 

1908 

108 

40 

40.42 

95 

28  35.44 

73 

31 

56.26 

338.546 

1912 

125 

49 

3.41 

95 

30  42.05 

73 

33 

10.74 

512.307 

1916 

143 

57 

26.41 

95 

32  48.64 

73 

34 

25.22 

86.068 

1920 

160 

5 

49.40 

95 

34  55.23 

73 

35 

39.72 

259.830 

1924 

177 

14 

12.40 

95 

37   1.81 

73 

36 

54.23 

433.591 

1928 

194 

23 

35.39 

95 

39   8.38 

73 

38 

8.75 

7.. 35  2 

1932 

211 

30 

58.39 

95 

41  14.94 

73 

39 

23.29 

181.113 

1930 

228 

39 

21.38 

95 

43  21.48 

73 

40 

37.84 

354.874 

1940 

245 

47 

44.38 

95 

45  28.02 

73 

41 

52.40 

628.635 

1944 

262 

56 

7.37 

95 

47  34.55 

73 

43 

6.97 

102.396 

1948 

280 

4 

30.36 

95 

49  41.07 

73 

44 

21.56 

276.158 

A  '1' 

1 

24 

28.007 

10.411 

6.100 

14.2715 

Factor  T 

4-0.222 

—.020 

+  .027 

—.0012 

A.'S 

+  .0007 

—.0001 

+.0001 

0 

THE   Oil  BIT   OF   URANUS. 


209 


TABLE  U.— Continued. 

2 

3 

4 

5 

6 

427 

7 

8 

9 

337 

1800 

r)ir).972 

513.802 

5 

511 

507 

591.1 

1804 

r)08.881 

527.803 

45 

524 

493 

582 

12.7 

417 

1808 

21.790 

641.805 

84 

538 

500 

50 

34.3 

497 

1812 

74.098 

555.806 

124 

.551 

20 

131 

.55.9 

577 

181(1 

127.007 

609.808 

103 

605 

93 

205 

77.5 

67 

1820 

180.510 

583.809 

202 

578 

159 

280 

99.2 

137 

1824 

233.424 

597.870 

242 

592 

225 

354 

120.8 

217 

1828 

2.S0.333 

11.872 

281 

0 

292 

429 

142.4 

297 

1832 

339.242 

25.874 

320 

19 

358 

503 

104.0 

377 

1830 

392.150 

39.875 

300 

32 

425 

678 

185.7 

457 

1840 

445.059 

53.870 

399 

40 

4  91 

52 

207.3 

537 

1844 

497.908 

07.878 

439 

59 

557 

127 

228.9 

17 

1848 

550.870 

81.879 

478 

73 

24 

201 

250.5 

97 

1802 

3.785 

95.880 

517 

87 

90 

270 

272.1 

177 

1850 

50.094 

109.882 

.557 

100 

157 

350 

293.8 

257 

iHfiO 

109.002 

123.883 

591! 

114 

223 

425 

315.4 

337 

18(!4 

li;2.51I 

137. S85 

35 

127 

290 

499 

337.0 

417 

1808 

215.420 

151.880 

75 

Ml 

350 

574 

358.0 

497 

1812 

208.328 

105.888 

114 

154 

( 23 

49 

380.3 

577 

isto 

321.237 

179,889 

153 

108 

489 

123 

401.9 

57 

1880 

374.140 

193.890 

193 

ISl 

555 

198 

423.5 

137 

1884  r 

427.054 

207.892 

232 

195 

22 

272 

445.1 

217 

1888 

479.903 

221.893 

271 

208 

88 

347 

400.7 

297 

1892 

532.872 

235.895 

311 

2:>2 

155 

421 

4X8.4 

37C 

1800 

585.780 

249.890 

350 

235 

221 

490 

510.0 

450 

11)00 

38.053 

203.889 

390 

249 

287 

570 

531.0 

530 

11)04 

91.502 

277.890 

429 

2i;2 

354 

45 

553.2 

16 

11)08 

144.470 

291.891 

408 

270 

420 

119 

574.9 

96 

1912 

197.379 

305.892 

508 

290 

487 

194 

590.5 

176 

1910 

2.50. 2S8 

319.894 

547 

303 

553 

208 

18.1 

256 

1920 

303.197 

333.895 

5S7 

317 

20 

343 

39.7 

336 

1924 

350.105 

347.897 

20 

3:!0 

80 

417 

01.4 

410 

1928 

409. OH 

301.898 

05 

341 

152 

492 

83.0 

496 

1932 

401.9.! 

-;..900 

105 

357 

219 

500 

104.0 

576 

1930 

514.831 

389.901 

144 

371 

2S5 

41 

i  120.2 

56 

1940 

507.740 

403.1^2 

184 

384 

352 

110 

147.8 

130 

1944 

20.049 

417.994 

223 

398 

41S 

1 90 

109.5 

216 

1948 

73.557 

431.905 
1.1500 

202 

411 

485 

205 
0.1 

,   191.1 
1.78 

296 
6.6 

4.3457 

3.2 

1.1 

5.4 

—  .0001 

-f  .0002 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

S 

7   June.  187 

3. 

210 


THE   ORBIT   OF   UIIANUS. 


TAI5LK  III. — IlEDiiCTioN  OF  THE  EpociiH  AND  Akqumentb  to  tue  beuinmnu  or  EACH  Month 

IN  A  Cycle  of  fouu  ykaus. 


YearO 

January 

Ft'bruary 

March 

April 

May 
June 
July 
Au^cust 


St'ptemljer  0 
October  0 
Xovcinber  0 
Ueccuiber    0 

Year! 

January 
Fi'liruary 
March 
April 

May 
Juno 
July 

August 


Scptoniber  0 
October  (» 
Nuvembor  0 
Ueceiuber    0 

Year  2 

January 
February 
March 
April 

May 

June 
July 
August 

September  0 
October  0 
November 
Ueceiuber 

Years 

January 
February 
March 
April 

Jfay 
June 
July 
Auirust 

S('|)teniber  0 
October  0 
November  0 
Ducembur  0 


9 


0  0  0.00 

0  21  411.24 

0  42  14.00 

1  4  3.24 


.  9 
10 
10 
II 

11 
11 
12 
12 


14 
14 
14 
15 

15 
10 
Hi 
IG 


2o 

4(> 

8 

2!) 

51 
12 

;54 

55 


10.24 

60. 4S 
0.48 
55.71 
44.95 
51.95 
41.19 
48.19 


4  17  37.42 

4  39  2(1.1)6 

4  59  9.19 

5  20  58.43 


42 

3 

25 

40 

8 
29 
51 
12 


59 

20 

41 

3 

25 
40 

8 
29 


15 

37 
58 
20 

42 
3 

25 
4U 


5.43 

54.07 

1.07 

50.90 

40.14 
47.14 
30.38 
43.38 


8  34  32.01 

8  50  21.85 

9  10  4.38 
9  37  53.02 


0.02 
49.80 
60.80 
40.09 

35.33 
42.33 
31.57 
38.57 


12  51  27.80 

13  13  17.04 
13  32  59.57 
13  54  48.81 


55.81 
45.05 
62.05 
41.28 

30.52 
37.52 
20.70 
33. 7C 


0  0  0.00 

0  0  2.09 

0  0  5.21 

0  0  7.89 

0  0  10.60 

0  0  13.19 

0  0  15.79 

0  0  18.48 

0  0  21.17 

0  0  23.77 

0  0  20.40 

0  0  29.06 

0  0  31.75 

0  0  34.44 

0  0  30.87 

0  0  39.56 

0  0  42.10 

0  0  44.85 

0  0  47.46 

0  0  50.14 

0  0  62.83 

0  0  55.44 

0  0  68.13 

0  1  0.73 

0  1  3.43 

0  I  6.11 

0  1  8.64 

0  1  11.23 

0 
0 
0 
0 

0 
0 
0 
0 


1  13.83 

1  l(i.52 

1  19.12 

1  21.81 

1  24.50 

1  27.10 

1  29.79 

1  32.40 


0  1  35.08 

0  1  37.77 

0  1  40.20 

0  1  42.89 

0  1  46.50 

0  1  48.18 

0  1  60.79 

0  1  63.48 

0  1  66.17 

0  I  68.77 

0  2  1.46 

0  2  4.06 


// 

0.00 
1.68 
3.05 
4.63 

6.15 

7.73 

9.25 

10.83 

0  0  12.40 

0  0  13.93 

0  0  16.61 

0  0  17.03 

0  0  18.01 

0  0  20.18 

0  0  21.61 

0  0  23.18 

0  0  24.71 

0  0  20.28 

0  0  27.81 

0  0  29.38 

0  0  30.96 

0  0  32.48 

0  0  34.00 

0  0  35.59 

0  0  37.16 

0  0  38.74 

0  0  40.10 

0  0  41.74 

0  0  43.20 

0  0  44.84 

0  0  40.30 

0  0  47.94 

0  0  49.52 

0  0  51.04 

0  0  52.62 

0  0  54.14 


0  1 

0  1 

0  1 

0  1 


0  55,72 

0  57.29 

0  68.72 

1  0.29 


1.82 
3.40 
4.92 
6.50 


0  1  8.07 

0  1  9.60 

0  1  11.17 

0  1  12.70 


& 


0.00 
0.00 
0.00 
O.Ol 

0.01 
0.01 
0.01 
0.01 

0.01 
0.02 
0.02 
0.02 

0.02 
0.02 
0.03 
0.03 

0.03 
0.03 
0.03 
0.04 

0.04 
0.04 
0.04 
0.04 

0.04 
0.05 
0.05 
0.05 

0.05 
0.(]^ 
0.06 
0.06 

0.00 
0.00 
0.00 
0.07 

0.07 
0.07 
0.07 
0.07 

0.08 
0.08 
0.08 
0.08 

0.08 
0.08 
0.09 
0.09 


Arg.  1 


0.000 

3.087 

7.136 

10.823 

14.391 

18.078 
21.647 
25.333 

29.019 
32.587 
30.274 
39.842 

43.529 
47.210 
50.640 
64.233 
57.801 
01.488 
65.056 
68.742 

72.429 
76.997 
79.0S4 
83.252 

86.939 
90.020 
93.956 
97.643 

101.211 
104.898 
108.400 
112.153 

115.840 
119.4(17 
123.094 
120.662 

130.349 
l:i4.o;i(! 
137.366 
141.053 

144.021 

148.308 
151.8(6 
155.563 

169.249 
102.S17 
100.504 
170.072 


TlIK   ORBIT   OP   URANUS. 


211 


TAULK  III 

—  Cuntinued. 

2 

3 

4 

5 

6 

7 

8 

9 

YearO 

January       0 

0.000 

0.000 

0 

0 

0 

0 

0.0 

0 

February    0 

1.123 

0.297 

1 

0 

1 

1 

0.5 

2 

March         0 

2.209 

0.575 

2 

1 

3 

3 

0.9 

3 

April  _         0 

3.25a 

0.872 

2 

1 

4 

4 

1.3 

5 

May            0 

4.382 

1.159 

3 

1 

5 

0 

1.8 

7 

June            U 

6.505 

1.457 

4 

1 

7 

7 

2.2 

8 

July            0 

C.5!)l 

1.745 

5 

2 

8 

9 

2.7 

10 

August        0 

7.714 

2.042 

0 

2 

10 

10 

3.1 

12 

Scptombcr  0 

8.837 

2.339 

7 

o 

11 

12 

3.0 

13 

October       0 

«J.y23 

2.02G 

7 

3 

12 

14 

4.0 

15 

November  0 

11.040 

2.923 

8 

3 

14 

15 

4.5 

17 

December    0 

12.132 

3.211 

9 

3 

15 

17 

4.9 

18 

Yearl 

January      0 

13.254 

3.508 

10 

3 

17 

19 

5.4 

20 

February     0 

14.377 

3.805 

11 

4 

18 

20 

6.9 

22 

March         0 

15.301 

4.073 

11 

4 

19 

22 

0.3 

23 

April          0 

1C.514 

4.370 

12 

4 

21 

23 

0.7 

25 

May            0 

17.000 

4.058 

13 

4 

22 

25 

7.2 

27 

June            0 

18.723 

4.955 

14 

5 

24 

27 

7.0 

28 

July             0 

10.809 

5.242 

15 

0 

25 

28 

8.1 

30 

August        0 

20.932 

6.539 

15 

5 

20 

30 

8.5 

32 

September  0 

22.054 

5.830 

10 

0 

28 

31 

9.0 

33 

October       0 

23.140 

0.124 

17 

0 

29 

33 

9.4 

35 

November   0 

24.203 

0.421 

18 

0 

30 

34 

9.9 

37 

Uecomber    0 

25.340 

0.709 

19 

0 

32 

30 

10.3 

38 

Year  2 

January       0 

21). 472 

7.000 

i^O 

1 

33 

37 

10.8 

40 

February     0 

27.505 

7.303 

20 

7 

34 

39 

11.3 

42 

March          0 

28.009 

7.571 

21 

7 

30 

40 

11.7 

43 

April           0 

29.732 

7.808 

22 

■y 

37 

42 

12.1 

45 

May             0 

30.818 

8.150 

23 

8 

3S 

44 

12.0 

47 

Juuo            0 

31.941 

8.453 

24 

8 

40 

45 

13.0 

48 

July             0 

33.027 

8.741 

25 

8 

41 

47 

13.5 

60 

August        0 

34.150 

9.038 

25 

9 

43 

48 

13.9 

52 

Sei)tembcr  0 

35.272 

9.335 

20 

0 

44 

50 

14.4 

53 

October       0 

3(1.358 

9.022 

27 

9 

45 

51 

14.8 

56 

November   0 

37.481 

9.919 

28 

<t 

47 

53 

15.3 

57 

December    0 

38.507 

10.207 

29 

10 

1 

48 

55 

15.7 

58 

Year  3 

1 

January       0 

39,090 

10.504 

30 

1       10 

50 

50 

10.2 

00 

February    0 

40.813 

10.801 

30 

10 

\       51 

58 

10.7 

02 

March          0 

41.827 

11.070 

31 

11 

1       52 

59 

17.1 

03 

April           0 

42.950 

11.307 

32 

'    11 

54 

01 

17.5 

.       05 

1 

May            0 

44.030 

11.0.54 

33 

1     11 

55 

02 

18.0 

i       07 

Juno            0 

45.159 

11.951 

34 

11 

57 

04 

18,4 

08 

July             0 

40.245 

12.239 

35 

12 

58 

05 

18.9 

70 

A  ugust        0 

47.308 

12.530 

35 

12 

69 

07 

19.3 

72 

September  0 

48.490 

12.833 

30 

12 

01 

08 

19.8 

73 

October      0 

49.570 

13.121 

37 

12 

02 

70 

20.2 

75 

Noveml)er  0 

50.099 

13.418 

38 

13 

04 

72 

20.7 

77 

December    0 

51.785 

13.705 

39 

13 

Oj 

73 

21.1 

78 

a 


}i 


^m 


212 


THK    OllUlT   OF    UKANLS. 


!r 


T.UILK  IV.— 

.Motion  oir  Auoiments  for  D.wa. 

Day». 

9 

u 

0 

1 

3 

3 

4 

5 

G 

7 

» 

9 

1     II 

II 

II 

1 

0  42.23 

0.09 

0.05 

0.119 

0.03(1 

0.010 

0 

0 

0 

u 

u 

0 

2 

1  24.47 

0.17 

0.10 

0  238 

0.072 

0.019 

0 

0 

0 

0 

0 

0 

3 

2     (■..70 

0.2(i 

0.15 

0.357 

0.109 

0.029 

0 

u 

0 

0 

0 

0 

4 

2  48.93 

0.35 

0.20 

0.470 

0.145 

0.039 

0 

u 

0 

0 

0.1 

0 

5 

3  31.17 

0.43 

0.2ft 

0.595 

0.181 

0.048 

0 

0 

0 

0 

0.1 

0 

f. 

4   13.40 

0.52 

0.30 

0.714 

0.217 

0.058 

0 

0 

0 

u 

0.1 

0 

7 

4  55. (;3 

0.01 

0.3(! 

0.833 

0.253 

0.007 

0 

0 

0 

0 

0.1 

0 

8 

5  37.87 

0.09 

0.41 

0.951 

0.290 

0.077 

0 

0 

0 

0 

0.1 

0 

!) 

(!  20.10 

0.78 

0  40 

1.070 

0.320 

0.08(5 

0 

u 

0 

0 

0.1 

0 

10 

7     2.33 

0.87 

0.51 

1.189 

0.302 

0.090 

0 

0 

U.l 

11 

7  44.57 

0.95 

0.50 

1.308 

0.398 

0.105 

0 

0 

1 

0.2 

12 

8  2(i.80 

1.04 

O.fil 

1.427 

0.434 

0.115 

0 

0 

] 

O.-J 

i;i 

9     9.03 

1.13 

O.fifi 

1.54G 

0.471 

0.125 

0 

0 

0.. 

14 

9  51.27 

1.21 

0.71 

1.C05 

0.507 

0.134 

0 

0 

* 

0.2 

15 

iO  33.50 

1.30 

0.70 

1.784 

0.543 

0.144 

0 

0 

1  :    1 

0.2 

Id 

11    15.73 

1.39 

0.81 

1.903 

0.579 

0.153 

0 

0 

1     1 

0.2 

17 

11   57.97 

1.47 

0.80 

2.022 

0.010 

0.163 

0 

0 

1 

0.2 

18 

12  40.20 

1.50 

0.91 

2.  Ml 

0.052 

0.173 

0 

0 

0.3 

19 

13  22.4.! 

1.(15 

0.97 

2.200 

0.C88 

0.182 

1 

0 

U.3 

20 

14     4.f.7 

1.74 

1.02 

2.378 

0.724 

0.193 

0 

^ 

0.3 

21 

14  4r,.90 

1.82 

1.07 

2.497 

0.700 

0.201 

0 

0.3 

22 

15  29.14 

1.91 

1.12 

2.filfi 

0.797 

0.211 

0 

0.3 

23 

1(1   11.37 

2.00 

1.17 

2.735 

0.833 

0.220 

0 

1 

0.3 

24 

K;  53. CO 

2.08 

1.22 

2.854 

0.8G9 

0.230 

0 

0.3 

25 

17  35.84 

2.17 

1.27 

2.973 

0.905 

0.240 

0 

0.4 

2(; 

18   18.07 

2.2(i 

1.32 

3.092 

0.941 

0.249 

0 

0.4 

27 

19     0.30 

2.34 

1.37 

3.211 

0.978 

0.259 

0 

1 

0.4 

2« 

19  42.54 

2.43 

1.42 

3.330 

1.014 

0.208 

0 

0.4 

29 

20  24.77 

2.52 

1.47 

3.449 

1.050 

0.278 

0 

0.4 

'■ 

30 

21     7.00 

2.(10 

1.52 

3.508 

1.080 

0.28S 

0 

0.4 

31 

21   49.24 

2.G9 

1.57 

3.087 

1.123 

0.297 

0 

0.5 

TAIU-I']  v.— MoTio.N  OF  <j  K(ia  Ilouits. 

Hours. 

.7 

lloura. 

.'/ 

Hours. 

V 

Hours. 

0 

// 

II 

II 

II 

0 

o.no 

fi 

10.50 

12 

21.12 

18 

31.07 

I 

1.70 

7 

12.32 

13 

22.88 

19 

33.43 

2 

3.52 

8 

14.08 

14 

24.04 

20 

35.19 

3 

5.28 

9 

15.84 

15 

20.40 

21 

3ii.95 

4 

7.04 

10 

17.(10 

IG 

28.10 

22 

38. 71 

5 

8,  so 

11 

19,30 

17 

29.92 

23 

40.47 

fi 

10.50 

12 

21.12 

18 

31.07 

24 

42,23 

The  period  of  nr{i;nments  1  to  9  is  600. 

Ill  January  and  Fi-liruiiry  of  thoso  years  which,  though  divisible  by  4,  are  not 
leap  years,  namely,  1700,  ISOO,  15)00,  2100,  etc.,  Table  IV  must  be  entered  with 
a  number  1  greater  than  the  real  day  of  the  month. 


TUE   OllBIT   OF   URANUS. 


213 


TAHLE    VI.— CullllKUTlONS  OK  AllUUMENTS  KOtt  TKIIM.S  (IF  LtiNll  PKlllnn. 


Year. 


u 


1000 
1010 
1U20 
1030 
1040 

lOJO 
lOUO 
1070 

lOSO 
lU'JO 

1100 
1110 
11^0 
1130 
1110 

1150 
IKiO 
1170 
IKSO 
11 'JO 

1200 
1210 
1220 
1230 
1210 

1250 
12U0 
1270 
12S0 
1200 

1300 
1310 
1320 
1330 
1340 

1350 
I3(i0 
1370 
13S0 
13'J0 

1400 
1410 
1420 
1430 
1440 

1450 
14(10 
1470 

14«0 
14!»0 

1500 
1510 
1520 
1530 
1540 


/       II 

+30  51.00 
3I>     (•..49 

35  22.14 

84  37. '.)7 

33  54.00 

+33  10.95 

32  2(;.71 

31  43.41 

31  0.37 

30  17.01 

+20  35.15 
2S  53.01 
2'<  11.20 
27  20.71 
2(i  48.5U 

+2ii  7.71 
25  27.27 
2t  47.1(1 

21  7.42 
23  28.00 

+22  40.0!) 

22  10.51 
21  32.31 
20  54.57 
20   17.23 

-10  40.31 

10  3.  S3 

is  27. so 

17  52.21 


+ 


17  17.0'J 

+  lfi  42.  n 

IC.  S.25 

15  34.55 

r-  1.31 

14  2S.(;3 

+  13  5r,.44 

13  24. 7(! 

12  53.  (;i 

12  22.00 

11  52.00 

+  11  23,35 

10  54.35 

10  25.02 

0  5.S.04 

9  30.74 


4.01 
37. SM 
12.34 
47.30 
23.01 


+   r.  59.20 

(;  311.14 
n  13.(11 

5  51.00 
+  5  30.40 


-0.014 
0.021 
0.020 
0.03S 
0.04S 

-0.0.50  — 
0.(172 
O.IISO 
0.702 
0.71S 

-0.735 
0.752 
0.770 
0.700 
0.800 

-0.820 

0.848 
0.800 
0.883 
0.899 

-0.914 
f     0.927 

0.039  , 

0.950 
I     0.000 

'—0.907 
'  0.972 
!  0.975 
!  0,977 
I     0.977  j 

'—0.974 
!  0.907 
I  0.059 
0.948 
j     0.030 

—0.021 
0,003 

!  0.8S3 
O.SOl 
0.837 

— O.SIO 
0.780 
(.748 
0.714 
0.078 

—0.041 
0.(101 
0.500 
0.518 
0.475 

—0.431 
().3S0 
0.340 
0.293 

—0.247 


-1.024 
1.003 
0.083 
0.902 
0.942 

■0.022 
0.902 
0.882 
0.802  1 
0.842  : 

-0.822 
0.803 
0.783 
0.704  ! 
0.745  I 

-0.720 
0.707  i 
0.089  j 
0,070  i 
0.052  I 

-0.034 
0.010 
0.598 
0,5S1 
0.5U4 

—0.547 
0.530 
0.513 
0.407 
0.4lil 

—0,405 
0.140 
0,433 
0,417 
0.402 

— 0.3S7 
0.372 
0.358 
0.344 
0.330 

—0.317 
0.303 
0.200 
0.277 

j     0.204 

—0.252 
0.240 
0.228 
0.210 
0.205 

—0.104 
j  0.1S3 
!  0.172 
I     0.102 

— 0. 153 


Year. 


+  1.702 
1.750 
1.720 
1.0S4 
1.040 

+  1.014 
1.570 
1.544 
1.500 
1.471 

+  1,430 
1 .  405 
1.370 
1.337 
1.304 

+  1.270 
1.237 
1.200 
1.172 
1.141 

+  1.110 

1.07  ■> 
1.040 
1.017 
0.980 

+0.957 
0.028 
0. 898 
O.S70 
0.842 

+0.814 
0.7  SO 
0.758 
0.730 
0.704 

+0.(;77 

0.051 
0.020 
0.002 
0.578 

+0.555 
0.530 
0.508 
0.485 
0.402 

+0.441 
0.420 
0.399 
0.378 
0.359 


1550 
1500 
1570 
1580 
1500 

1000 
1010 
1020 
1030 
1040 

1050 
1000 
1070 
lOsO 
1090 

1700 
1710 
1720 
1730 
1710 

1750 
171.0 
1770 
1780 
1700 

1800 
181)1 
1802 
1803 
1801 

1805 
1800 
1807 
180S 
1809 

1810 
1811 
1812 
1813 
1814 

1815 
1810 
1817 
1818 
1819 

1820 
1821 
1822 
1823 
1824 


+5 
4 
4 
4 
3 


II 

9.73 
49.70 
30.:!2 
11.57 
53.48 


+3  30,03 
3  19.20 
3     3.14 

2  47.09 
2  32.89 

+  2  18.77 


1 
1 
1 

-fl 
1 
0 
0 
0 


52.55 
40.40 
29.05 

18.32 
8.28 
58.92 
50.24 
42.2(1 


+  0 
0 
0 
0 

+0 

+8!'84 
8.49 
8.15 
7,82 
7,50 

+7.18 
0.87 
0.57 
0.27 
6. 


31.90 
28.30 
22,44 
17.22 

12.08 


—  jj 
•34 

•33 

•32 

—  31 
•30 
•30 

.2<) 

•  2y 

+5.00  _ 
5.41      '^'^ 
5.14     --7 

4.88     -^^ 
4.02     -'^ 


.98 
0 


+0,340 

1825 

0,322 

1820 

0,303 

1827 

0,284 

1828 

+0.208 

1829 

+4.37 
4,13' 
3,s9 
3,00 
3.44 

+  3.23_ 

3.02  • 
2. 82     ■ 

2.03  ■ 
2.44     • 

+2.20_ 
2.09 
1,92 
1,70 

j  +  1.01 


.24 

.24 

•23 
.22 
.21 


—0.200 
0,153 

0,1 (h; 

0.000 
—0,013 

+  0.034 
0.078 
0,120 
0.102 
0.203 

+0,242 

0,278 
0.312 
0.345 
0,375 

fO,403 
0,429 
0,452 
0,472 
0,489 

+0,503 
0,514  I 
0,522 
0,527 
0,520 

+0,528 
0,528 
0,528 
0,527 
0.527 

+0,527 
0.520 
0.520 
0.525 
0,525 

+  0,524 
0,523 
0,523 
0.522 
0.522 

+  0  521 
0.520 
0.520 
0.510 
0,518 


17 
■»7 
.16 

•■5 
—  li 


+  0.517 
0.51(1 
0.515 
0.514 
0.513 

+0.512 
I  0.511 
I  0.510 
i     0.509 

—0.508 


-0.144 
0.134 
0.125 
0.117 

0. 108 

-0.100 
0.092 
0.085 
0.078 
0,071 

-0,004 
0.058 
0.052 
0,040 
0,041 

-0,030 
0,031 
0,027 
0.023 
0,019 

-0,015 
0.012 
0.009 
0.007 
0.005 

-0,004 
0,004 
0.004 
0.004 
0,004 

—0,003 
0,003 
0.003 
0.003 
0.003 

—0.003 
0.003 
0.003 
0,003 
0.002 

-0.002 
0,002 
0,002 
0,002 
0,002 

-0,002 
0.002 
0.002 
0.002 
0.002 

-0.001 
0.001 
0.001 
0.001 

-0.001 


+  0.252 
0.230 
0.210 
0.205 
0.189 

+  0.175 
0.101 
0.148 
0.135 
0.123 

+0.112 
O.lOl 
0.091 
0.081 
0.072 

+  0.003 
0.054 
0.040 
0.030 
0.032 

+  0.020 
0.021 
0,017 
0.013 
0,010 

+0,007 
0,007 
0,007 
0,007 
0,007 

+  0,000 
0,000 
0,000 
0.000 
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+0.005 
0.005 
0  005 
0.005 
0.005 

+  0.004 
0.004 
0.004 
0.004 
0.004 

+  0.003 
0.003 
0.003 
0.003 
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+0.002 
0.002 
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214 


TUE   ORBIT  OF    URANUS. 


TAULIO   y\.—(!i>iilii,ti,il. 


Toar. 


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1831 
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1833 
1834 

1835 
1830 
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1838 
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1840 
1841 
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1844 

1845 
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1850 
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1854 

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1857 
1858 
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18()3 
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1805 
1800 

1807 
1808 
1809 

1770 
1871 
1872 
1873 
1874 

1875 
1870 

1877 
1878 
1879 

1880 
1881 
1882 
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1884 


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+  .20 
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0.500 
0.505 
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0.500 
0.498 
0.497 
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+0.494 
0.493 
0.491 
0.490 
0.488 

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0.485 
0.483 
0.482 
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0.477 
0.470 
0.474 
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0.409 
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0.405 
0.403 

+0.401 
0.459 
0.457 
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+0.451 
0.449 
0.440 
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1 885 

1880 
1887 
18S8 
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18111 
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1895 
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2120 
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0,155 
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+  0,182 
0,190 
0,210 
0,225 
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+0,250 
0,273 
0,289 
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TllK   OR  HIT   OF   UIIANU8. 


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3(1211 

254 
256 

50' 

10' 

3     0  37.73 

49.10 

64(113 

355 

50' 

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54-54 

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40 

20 

3     1   2(1,73 

49  00 

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357 

40 

ao     2   10  2'.i.22 

40    2  11  2a.i;o 

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a(l98i; 

259 
260 

30 

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30 
40 

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3     a     4.41 

48. 89 
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55329 
55(189 

359 

300 

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30 
20 

60     2   12  17. 'JO 

1 

54-30 
54- -i^ 

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263 
263 

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50 

3     3  5,!. Oil 

48. 68 
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5(1050 

10 

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10'    2   U     (l.-.M) 
20     2   15     0.a2 

54-14 
54-06 

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38043 

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267 

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40 

33 

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20 

3     4   41.(1(1 
3     5  .'Id.  12 
3     (1    18.47 

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364 
366 

367 

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40 

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3     7      (172 

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48.13 

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40     2   1(1  4H,20 

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3   10    is., -,8 

47.80 
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30 

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30 
40 
50 

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390 

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30 
20 
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42550 

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52.62 

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3    19    43.31 

46.43 

03592 

392 

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52-53 

43144 

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40 

2() 

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46.32 
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03986 

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40 

30     2  31   51.(10 

52-44 

43444 

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30 

30 

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40 

2  32   43.95 

52-31 

4. 17  45 

301 

20 

40 

3  22      1.91 

46.08 

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396 

20 

50 

2  33  311.21 

52.26 
52.17 

44048 

303 
305 

10 

50 

3  22  47.88 

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397 
399 

10 

27° 

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2  a.')  •'(»  4il 

52. oS 

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44(159 

306 

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10 

3  23  3:!.  72 
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400 

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51.98 

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308 

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20 

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407 

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28°;  ■:;  39  39.411 

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51-52 

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51.42 

4(1850 

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20 

3   29  3(1.13 

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30     2  42   13.72 
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5 '-32 
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47170 
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320 
321 

30 
20 

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44-77 
44.64 

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20 

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5'-'3 

47814 

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50 

3  31    50.05 

44-52 

70047 

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416 

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1 

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324 

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3  32   3!. 4 5 

70403 

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5094 

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326 

50' 

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44-27 

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50.84 

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20 

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44.14 

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30      2  47    19.tl4 

50-74 

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330 

30 

30 

3   34   4(1.88 

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43-77 
4364 

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1 

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335 

424 

30'    2  49  51.28 

i 

50121 
1.26 

330° 

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40 

3  3fi  58.19 

72984 
1.26 

320° 

THE   OlllJIT    OF   UllANUS. 


317 


TAIILK  VII. 

—C, 

liliinii'd. 

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r 



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1.27 

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803.87 

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446 
448 

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10 

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33  0° 
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10 

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40  4 !.!!(! 

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50 

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463 

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50 

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523 

10 

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57   47.47 

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30 

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17595          « 
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30 

40 

4 

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39. 17 

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466 

20 

40 

4  35 

13.13 

30.42 

20 

50 

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1      4.01 

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50 

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30.27 
30.12 

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10 

46° 

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13.53 

29.96 

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2  21.(15 

3'^-75 
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89.5(14 

471 

50' 

10' 

4  3(1 

43.48 

19712     530 

50' 

20 

4 

3     0.27 

90035 

47' 

40 

20 

4  37 

13.29 

29. Si 
29.65 

20243     53' 

40 

30 

4 

3  3S.75 

3,S.4,S 

90507 

472 

30 

30 

4  37 

42.04 

20770     ^^•? 

30 

40 

4 

4   17.00 

3''^-3  4 

909S0 

473 

20 

40 

4  38 

12.43 

29.49 

213(10     \\\ 

20 

50 

4 

4   55.30 

3.S.21 
3S.06 

91455 

475 
476 

10 

50 

4  38 

4'. 77 

29-34 
29.19 

21842     5-33 

10 

47° 

4 

5  33. 3r, 

91931 

313' 

57° 

4  39 

10.00 

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303° 

10' 

4 

0   11.20 

37-93 

92408 

477 

50' 

10' 

4  39 

30.00 

29.03 

22011     533 

50' 

20 

4 

0  40.07 

37- 7« 
3-65 
37-50 
37-36 
37- -'2 

92SS6 

47S 

40 

20 

4  40 

8.87 

28.88 
28.72 
2S57 

23448     537 
23080     "° 
24524     53« 

40 

30 
40 

4 
4 

7  2n.72 
S     4.22 

933(15 
93S45 

479 
4S0 

30 
20 

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40 

4  40 
4  41 

37.50 
0.10 

30 
20 

50 

4 

8  41. 5S 

0432T 

4,82 
4S3 

10 

60 

4  41 

34.57 

20.41 
2S.26 

25003     53^ 

10 

48° 

4 

0   IS. 80 

94SI0 

312° 

58' 

4  42 

2.83 

2S.10 

25003 

302° 

10' 

4 

9  55. SS 

37-o.S 

95294 

4S4 

60' 

10' 

4  42 

30.03 

20143     540 

50' 

20 

4 

10  32.. Si 

36-93 

95779 

4«5 

40 

20 

4  42 

58.87 

27-94 

20084     54' 

40 

30 

4 

11      9.(10 

36-  79 

0112(15 

486 

30 

30 

4  43 

20.0(1 

27-79 

27220     542 

30 

40 

4 

11   40.25 

36-65 
36.51 

9(1752 

4.S7 

20 

40 

4  43 

54.29 

27.63 

27708     542 

20 

50 

4 

12  22.7(1 

97240 

488 

10 

50 

4  44 

21.70 

27--I7 

28311     543 

10 

36-36 

49r 

27-3' 

544 

49° 

4 

12  50.12 

36.22 

07730 

311° 

59° 

4  44 

49.07 

27.16 

2S8.»,5     ^  , , 

301° 

10' 

4 

ly  35.34 

98220 

490 

50' 

10' 

4  45 

10.23 

20399     544 

50' 

20 

4 

14   11.41 

36.07 

98711 

49' 

40 

20 

4  45 

43.22 

26.99 
26.84 
^6.69 

20044     545 

40 

30 

4 

14  47.34 

35-93 

00203 

492 

30 

30 

4  4(1 

10.00 

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30 

40 

4 

15  23.13 

35-79 

99(190 

493 

20 

40 

4  4f. 

30.75 

31038     547 

20 

50 

4 

15  58.77 

35-64 
35-5° 

*(H)101 

495 
496 

10 

50 

4  47 

3.27 

21'.  5  J 
'6.37 

31587     549 

550 

10 
300° 

50° 

i 

Ifi  34.27 

*00fl87 

310° 

60' 

4  47 

29.04 

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l_ 

♦1.27 

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1.27 

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1^' 


28       June,  1873. 


I  : 


I 


i 


i 


:  t 
.    :  f 


'21S 


TM  K   OIMIIT  O  i'    U  II  A  N  US. 


rAUl.K   Vll 

—  Continued. 

9 

h' 

1.527 

r 

<.l 

0   1   II 

II 

l.'it;.  r 
1.27 

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II 

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4 

4T   'J!U!4 

iCt.  2  1 

;;ji;i7 

300 

70° 

ft     s  ft7.0l 

6.47 

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290'^ 

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4  7  ftft.Hft 

;fj(is7 

f>0' 

10' 

ft    0  l;!.4s 

cc7:!s      ^ 
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fto' 

L'd 

1 

4S  J  1.00 

3(».o5 

;{;tj;i7 

ceo 

40 

JO 

ft    0  JO.  70 

40 

;io 

1 

4S  47.70 

25., Si, 

:i:t7H7 

.S.SO 

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:io 

ft     it  4ft. on 

6.14 

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40 

4 

40   l;f.ft.'J 

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ft    10      I.OI 

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2S4J 

;j4syo 

5. Si 
55J 

10 

ftO 

ft  10   17.72 

5.  Si 
5- ('4 

"■"'•'  ,'i:,i 

10 

er 

i 

ftO     4.5'i 

25,  2(i 

;}ft443 

209° 

71° 

ft   10  WW.-M', 

5-4H 

SS  >;; 

289° 

10' 

4 

ftO  JO. 78 

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S.S.? 

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10' 

ft   10  4S.S4 

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20 

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4-oS 
4. Si 

4. ''5 

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4 

ftl    4  (.CO 

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4 

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2.;  Si 
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559 

JO 

40 

ft  i:i    o.iio 

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7ftftll       • 

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4 

ft;  :i;i  ftc. 

4ift(;5 

S''o 

10 

ftO 

ft  i;i  14. ftl 

7CO0S    ■;'„' 

10 

2 J  49 

5<" 

J.  06 

5'^7 

63' 

4 

ftj  ft7.0ft 

4JP2r. 

297' 

73 

ft  i;i  2S.I7 

7C«sft 
77J7:i    -'^'^ 

28T 

10' 

4 

ftft  -JO  ;is 

2,mS 

4J(!S7 

5f>' 
5<'>-' 

fto' 

10' 

ft  i:{  4i.('.i; 

.?-49 

fto' 

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4 

ftft  4;i.ft« 

4:!J10 

40 

20 

ft  i:t  ft 4. OS 

.5-.!2 

77SC1      ■^''^ 

40 

;io 

4 

ftO    (i.ft7 

2,?. 01 

4:1s  11 

S^''' 

;!0 

:!0 

ft  14    s.l:t 

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7si.fto    v;;9 

;)o 

40 

I 

ftC)  jo.4'a 

22..S5 

44:174 

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JO 

40 

ft  14  JI.12 

2-99 

7oo;(o    ^^J> 
70.IJS    5^'^ 

JO 

;.0 

4 

ftC.  ftJ.  1 1 

2  2 .  f  )c) 

440;i3 

5 ''4 

lU 

i)0 

ft    14   :t.!.04 

2.S2 

10 

^^  ii 

5''5 

2.66 

5''<9 

64° 

4 

ft7   M.(M 

4ftfto:i 

296" 

74 

ft   14   40. CO 

.SO  J 17 

286' 

10' 

4 

ft7  :i7.01 

2'-.W 

4i;oiis 

5^>5 

fto' 

10' 

ft    It   ftO.OO 

-'•49 

SOS07     -''''° 

fto' 

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4 

ft7  ftO.Jl 

22.  20 

4(!C.:!:i 

SH 

40 

'JO 

ft  1ft  11  (1 

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si;:'.i7     ■'^'^" 

40 

:!o 

I 

fts  JI.-J5 

22.04 

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5(>f> 

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ft  Ift  J:t.ft7 

2,16 

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:io 

40 

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fts  4:!.i:t 

2  1  .  SS 

477(ir. 

5 ''7 

JO 

40 

ft  ift  aft.ftc 

1.1)1) 

sjft77    l'>° 

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rio 

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21.71 
21.56 

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10 

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ft  Ift  47.;io 

l.S,5 
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10 

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295 

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ft  Ift  ft 0. Oft 

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40 

JO 

ft  n;  JI.S7 

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sioil     5''" 

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0  :!o.oo 

21.07 

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ft  1<;  :i:!.o:i 

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SftftllJ      •^'" 

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40 

ft 

0   ftO.OO 

20.  ()0 

ftl  177 

57" 

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40 

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1.00 

scut    •''''•• 

JO 

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ft 

1  ii.7;t 

20.7.1 
20.58 

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57" 
57' 

10 

fto 

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s,;:ift    ■'^'" 

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214' 

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1  ft  J.  7 -J 

20.41 

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57' 

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10 

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0.50 

s.soo      "^ 

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ft 

J   1J.07 

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40 

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ft  1  cor. 

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40 

ft  !7   4C,ftl 

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r.o 

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571 

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10 

67 

ft 

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293' 

77 

ft  IS    c.oo 

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283' 

10' 

ft 

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")-4.? 

ft(',:iJ7 

574 

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9-50 

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4   11.04 

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571 

40 

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10 

40 

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ft    IS   ftl. SI 

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S.S,5 
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JO 

10 

68'  :. 

ft  jr.  (!) 

ft  0 -JO  4 

292' 

78 

ft   10     tJ,  (7 

8.^0 

0(4J( 

282' 

10    r. 

r.  4  1.04 

1S.45 

ft  07  SI 

577 

fto' 

10' 

ft    10     S.07 

OftOIS     594 

fto' 

JO    :> 

c,    ;t.j:t 

IS.JCJ 

18.12 

(io:!ftS 

577 

40 

JO 

ft  10  I7.:!0 

S-.l.i 
S.  1 6 

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7\f 
7,66 

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40 

;iO 

ft 

c.  Ji.;t5 

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57'^ 

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ft  10  jft.4n 

OC.JOC     594 

liO 

40 

ft 

fi  ;io.;u) 

«7')5 

f.  1  ft  1 4 

57''< 

'JO 

40 

ft  10  .'i:i.4C 

ocsoo    594 

JO 

TiO 

ft 

(i  ft7.09 

>7"9 

iijoo:} 

570 

10 

ftO 

ft   10  4 1. JO 

ii7;!0ft    505 

10 

'763 

570 

505 

69'  r. 

7  14.7^3 

17.46 

r,jfi7:i 

291° 

79 

ft   10  4S.0ft 

07000 

281° 

10'   :. 

7  :iJ.ls 

(i.'i'jftj 

5"') 

ftO' 

lit' 

ft    10  ft C.  4ft 

7-5" 

Osftsft     595 

fto' 

JO     Ti 

7    40. 4S 

>7-.'>o 

(;;!s:!j 

5  So 

40 

JO 

ft   !20     .•t.7S 

1-.n 

00170     594 

40 

;io    ft 

«      fl.Ol 

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(U41J 

5''o 

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:!0 

ft   JO    10.01 

7.16 

00771    50.-; 

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40    r. 

s  j:i.ft7 

16  ()6 
16.  So 

(!400:! 

5S1 

JO 

40 

ft   JO   17.01 

7.00 
O.S., 

♦oo:!t;o    5'b 
♦OOOCft     sofi 

JO 

ftO     ft 

S   40., '17 

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10 

ftO 

ft   •JO  J (.77 

10 

1 6. 6  ( 

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1 

8  ft7.01 

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1.27 

290 

80 

ft  JO  ;ii.44 

♦iMftr.o 
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280° 

7 

T  H  K   ORBIT  OF   URANUS 


219 


TAIU.K   \'\\—<hmlinwd. 


80" 

1(1' 
20 
30 

4U 
50 

81" 

10' 
20 

no 

40 
f>0 

82° 

10' 

20 

:io 

40 
50 

83° 

10' 

20 

:io 

40 
50 

84' 

10' 

.'id 

40 
50 

85° 

10' 
20 
•M 
40 
50 

8G" 

Id' 
20 

;io 

40 
5() 

87° 
10' 
20 

:io 

40 

50 

88° 

10' 
20 

:to 
40 
50 

89^ 

10' 

•io 
no 

40 
50 

80' 


20 
20 
20 
20 
20 
21 

21 
21 
21 
21 
21 
21 

21 
21 
21 
21 
21 
21 


:tl.44 

:n.ii4 

41.2S 

50.45 

5(!.4f! 

2.:io 

l.'.ir, 
i:i.4<> 

IH.SO 
2:t.!>7 

2H.1I7 

n.f.sl 
;is.4s 

42.'.lil 
47.:t:t 

51,50 
55.51 
511, :!0 

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(;.5r, 

11. 01 

I  :;.()'.> 
ii;.il 
is,'.)7 

21.r,(', 
21, 10 
2l'i,.i5 
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lis,  1  1 
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40.:i5 

41.  on 

4  I.  (SO 
22  II. '.I'.) 
22  42.21 
22  12. 2i; 
22    12.15 

22  41.SS 

J.»  41.15 

5   -zi  40. S5 

5  22  4<».()0 

5  22  :i'.(.  Ii; 

5  22  :!s.07 


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22 


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6. 

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6. 

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6 

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6 

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5 

66 

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5 

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4 

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5  22  ;!5.I0 

5  22  ;!:i.H2 

5  22  ;i2 OS 

5  22  :!0. 17 

5  22  2S.I0 

5  22  25. H(! 

5  22  2:1.17 

5  22  20.1)1 

5  22   IS.  ID 

5  22   15.;tl 

5  22   12. 2(! 

1  5  22     0.05 


4-5" 
4  ,il 
4  '7 
.).ol 

.V«.S 
.,.68 

.V.i.S 
.,.i,S 

3  "- 

2..S6 
2.6y 

2.y, 

2.20 
2.05 

I..S(> 
1.70 

'-.S.i 
'  ,57 
1 .  20 
I  o( 
0..S7 
0.71 

0-.S4 
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0.05 
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0.-7 

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o.Oo 
0.76 

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1 .01; 

I.  t2 

i,5,S 

1  74 

2.07 
2.2.t 

2  ,V/ 
2.56 
2.72 
2.SS 


1.28 

ol5<;o 

02155 
02750 

o:!:i4<> 

OltlMl 
OI5:ifl 

05i:tl 

0572(1 
0(i:i22 
0(1017 
07512 
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0^702 
((0207 

00sii;i 

KIIMS 

IHw:t 
I  Kill) 

1 227  I 
1 2-ti;i» 
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1  KICO 
1  1(155 
15250 

I5S4(! 
1(1111 
170.111 
17(i:il 
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IS.S20 

10115 

200 1 II 
201IOI 
21  I  OH 
21702 
22:!SI1 

22its0 
2:157  I 
211(17 
217(11 
25:!5I 
2511 H 

2(1511 
■.;7l:!l 
27  7211 
2s:i!ll 
2S1)II 
21150 1 

:s(ioi)(> 
iiodss 

:!12S0 
:il.s72 
:i24(i4 
;;::o55 

:!:!(!  »7 
ill  2:1s 
;!IS2H 
:!54I.S 

:ti'(i(m 
:t(;5ii7 

:ni^7 
123 


SO.S 
5V.S 
5'/' 
S'JS 
50.S 
595 

595 
5'/> 
505 
595 
595 
595 

595 
59^' 
595 
595 
59^' 
595 

595 
595 
595 
595 
595 
59^' 

595 
595 
595 
595 
59  1 
595 

595 
591 
59  1 
59  1 
591 
59  1 

594 
59.5 
59  1 
59.1 
591 
59.1 

$')' 
59.! 
59.: 
59.1 
59-! 

592 
592 
592 
59-' 
59" 
592 

59' 
59" 

590 

5''<9 

590 


280 

50' 
40 
.•10 
20 
lU 

279' 

50' 
40 

:iO 

20 
10 

278 

50' 
40 
30 
20 
10 

277' 

50' 
III 
30 
20 
10 

276 

50' 
40 
:!0 
20 
10 

275 

50' 
40 
30 
20 
10 

274 

50' 
40 

:to 
20 

10 

273 

5ii' 
4(1 

:io 
20 
10 

272 

50' 
40 
30 
20 
(0 

27  V' 

50' 
40 

:;o 
20 
10 

270 


K 


90" 

10' 

20 
30 
40 
50 

9r 

10' 
20 
30 
40 
50 

02" 

10' 

20 
:to 
40 
50 

93' 

10' 
20 
30 
40 
50 

94' 

10' 

20 

30 

40 

50 

95" 

10' 

20 

;io 
to 

50 
96° 

10' 
20 
30 
40 
50 

07° 

10' 
20 

30 
40 
5(1 

08° 

111' 
20 
30 
40 
50 

99' 

10' 
20 
30 
40 

50 

100 


i  22 
I  22 
>  22 
1  21 


5  21 
5  21 

5  21 
5  21 
5  21 
5  21 
5  21 
5  21 

5  21 
5  21 
5  21 
5  21 
5  20 
5  20 

5  20 
5  20 
5  20 
5  20 
5  20 
5  20 

5  20 

5  111 

5  10 

5  111 

5  10 

5  10 

5  10 

5  10 

5  10 

5  IS 

5  IS 

5  l.S 

5  IS 

5  IS 

5  IS 

5  17 

5  17 

5  17 

5  17 
5  17 
5   17 


5   111 

5  1(1 

5  1(1 

5  111 

5  15 

5  15 

5  15 

5  15 
5  15 
5  II 
5  I  I 
5  II 
5    II 

5    It 


0.05 

5.(1H 

2.15 

5S.45 

51.50 

50. 5H 

4(1.40 
42.0(1 
37.57 
32.01 

2S.01I 
23.11 

17.07 

12.(17 

7.20 

I.5S 
55.70 
4  0.S5 

43.74 
37.4  s 
;!I.(I5 
24.17 
17.72 
10. H2 

3.7(1 
5(1.51 
40.1(1 
41.(12 
3:!.o:t 
211.07 

IS.  0(1 

O.sil 

1.5(1 

5:!. (Id 

41.42 

35.(11 

211.115 
17.53 
S.2I1 
5s.  S3 
40.24 
311.. 50 

211  (10 

10.5! 

Ii.:i:t 

5S.1M1 

4s. 44 
.■!7.7(1 

2(1.02 
15.03 
4.7s 
5:!.  IS 
42.(12 
30.41 

IS. (11 
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51.111 
4  2.41 
:io  11:1 
17.111 

4.  so 


,r.i7 

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3-7° 
3.X6 
4.01 
4.1.S 

4. 54 
4-49 
4.66 

4.«2 

4.9H 
5- '4 

5  •,■50 

5'47 
5.62 

5-79 
5'9l 
6.11 

6.26 

(-.4,5 

6.5,S 

6.90 
7.o(j 

7.22 
7-.5« 
7-54 
7.6.) 
7..S6 
■S.oi 

K.I  7 
«..5,5 
8.50 
8.6.) 
8.81 
8.(/) 


12 

7 
4.5 
59 
71 
90 
10.06 
io  2 r 

10.  -,7 
10.52 
10.68 
I0.8.I 

10. ()9 
11.15 

• '  .50 
I  l..|6 
11.61 
11.77 

I  1.92 

12.08 

12  2.1 
I2..?8 
12.5.1 
I  2.69 


1.28 

37187 
:!777(l 
3S3(!5 
38051 
311:43 
40131 

40710 
41307 
41H04 
424S2 
4:10(10 
43(1.55 

44242 

44S28 
45114 
4(1000 

4I15S.^. 
47170 

4  7755 
4s:i;{0 
4sll2:i 
40507 
500110 
501173 

51250 
5Is:js 
52420 
5:tO(l2 

5:f5s:i 
541(13 

54743 
55:12:1 
55II02 
5(14  Si 
570(10 
571138 

5821(1 
5M71I4 
50371 
5004S 
(10525 
(IIIOI 

(11(177 
(12252 
H2S27 
ii:t4(il 
(i:i075 
(14548 

(15120 
(1511112 
(1112(14 
(ir,s:tr, 
(17 107 
(17077 

(1S547 
110117 
(loiisii 
711255 
7IIS23 
71301 

71058 

i?.t; 


270" 


5«9 

50 

5«9 

40 

5«9 

30 

5«9 

20 

588 

10 

5. 88 

268' 

58S 

bU 

5«7 

40 

5,88 

30 

5«7 

20 

586 

10 

5«7 

2«H' 

5  86 

50' 

586 

40 

586 

30 

S«5 

2U 

5«5 

10 

S«5 

267^ 

5'^4 

5(1' 

S«4 

41» 

5«4 

30 

5«.5 

20 

SH.5 

10 

S«1 

266" 

S«2 

50' 

5«2 

40 

S''^-' 

30 

5«' 

20 

580 

10 

5,80 

5  So 
579 
579 
579 
57« 
57« 

573 
577 
577 
577 
57'-' 
S7'-> 

575 
575 
571 
574 
57.5 
57-> 

572 
572 

572 
57' 
570 
570 

570 
5'>9 
$(>') 
568 
568 
5 ''7 


,  1  -i 


265° 

50' 
40 
30 
20 
10 

264° 

50' 
40 
30 
20 
10 

263" 

50' 
40 
30 
20 
10 

262" 

5(1' 
40 
30 
20 
10 

261" 

.50' 
40 
30 
20 
Id 

200 


220 


THE   ORBIT   OF   URANUS. 


TABLE  \\l. -Continued. 


E 


100° 

10' 
20 
30 
40 
50 

lor 

10' 
20 

;to 

40 
50 

102' 

10' 
20 
30 
40 

f)0 

103° 

10' 
20 
30 
40 
SO 

104° 

lu' 
20 
30 
40 
50 

105° 

10' 
20 
30 
40 
50 

106° 

10' 
20 
30 
40 

50 

107° 

10' 

20 

30 

40 

50. 

108° 

10' 
20 
80 
40 
50 

109° 

10' 
20 
30 
40 
50 

UO^ 


14  4.80 
13  51.96 
13  38.96 
13  25.81 
13  12.51 
12  59.06 

12  45.45 
12  31.(19 
12  17.78 
12  3.72 
11  49.50 
11  35.14 


5  11  20.62 

5  11  5.95 

5  10  51.13 

5   10  36.16 

5  10  21.04 

5  10  5.77 

5     9  50.35 

9  34.78 

9  19.06 

9  3.19 

8  47.17 

8  31.01 


14.69 

58.22 
41.61 
24.85 
7.94 
50.88 


6  33.67 
6  16.32 
5  58.82 
5  41.17 
5  23.37 
5     5.42 


5  0 

6  0 
5  0 
4  .'■)9 
4  59 
4  59 

4  .^S 
4  5S 
4  58 
4  57 
4  57 
4  57 


47.33 
29.10 
10. -:i 
!)2.19 
33.52 
14.70 

55.74 
3(>.63 
17.38 
57.99 
38.45 
18.77 

58.94 
38.97 
18.H6 
5  s.  60 
38.21 
17.67 

56.98 
36.15 
15.18 
54.07 
32.81 
11.42 


4  56  49.88 


2.84 
3.00 
3»5 

,?-45 
3.61 

,V7<5 

4.06 
4.22 

4- 36 
4-52 

4.67 
4.S2 
4-97 
5->- 
5-27 
5-42 

5-57 
5-72 
5-87 
6.02 
6.16 
6.32 

6.47 
0.61 
6,76 
6.91 
7.06 
7.21 

7-3S 
7-5° 
7-6.S 
7.  So 

7-95 
S.09 

S23 
8.38 

8.53 
8.67 
8.82 
8.96 

9.11 

9-25 
9-39 
9-54 
9.68 

9-83 

19.97 
20. 1 1 
20.26 

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20.69 

20.83 
20.97 
21. 1 1 
21.26 

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1.23 

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72525 
73091 
73656 
74221 
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75348 
75911 
76474 
77036 
77598 
78159 

78720 
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79839 
80398 
80956 
81514 

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82628 
83184 
83739 
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88160 

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89258 
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90354 
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94172 
94715 

95257 
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96338 
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565 
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553 
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549 

549 
548 
548 
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547 
546 

545 
545 
544 
543 
543 
542 

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540 
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539 
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538 

537 
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535 
535 
534 
533 

533 

532 

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530 

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1/ 

260' 

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50' 

10' 

40 

20 

30 

30 

20 

40 

10 

60 

259° 

111° 

50' 

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40 

20 

30 

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20 

40 

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258' 

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50' 

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40 

20 

30 

30 

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20 

30 

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114° 

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20 

30 

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54  37.72 
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53  52.55 
53  29.76 
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4  53 

4  52  43.76 


52  20.55 
51  57.20 
51  33.72 
51  10.10 
50  46.34 
50  22.44 

49  58.41 
49  34.24 
49  9.94 
48  45.50 
48  20.92 
47  66.21 

47  31.36 

47  6.38 

46  41.26 

46  16.00 

4  5  50.61 

45  25.09 

44  59.44 
44  33.66 
44  7.74 
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43  15.50 
42  49.19 

42  22.74 
41  56.16 
41  29,45 
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40  35.62 
40  8.52 

39  41.28 
39  13.91 
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THE    OllBIT   OF   UK  AN  US. 


221 


TAULK   Vn.—Conlinued. 


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10' 
20 
30 
40 
50 

121" 

10' 
20 
30 
40 
50 

122' 

10' 
20 
30 
40 
50 

123' 

10' 
20 
30 
40 
50 

124° 

10' 
20 
30 
40 
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125° 

10' 
20 
30 
40 
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31  !).01 
30  39.35 
30  9.56 
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29  9.(il 
28  39.45 

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27  38.75 

27  8.22 

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25  35.88 

25  4. 85 
24  33.71 
24  2.44 
23  31.05 
22  59.54 
22  27.91 

4  21  50.10 

4  21  24.29 

4  20  52.29 

4  20  20.18 

4  19  47.94 

4  19  15.59 


126° 

12 

4.35 

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30.37 

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30 

4 

10 

22.00 

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9 

47.73 

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20 

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54.40 

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20 
30 
40 
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129' 

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20 
30 
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4  14  52.55 

4  14  19.14 

4  13  45.02 

4  13  11.98 

4  12  3S.22 


3  58 


9.00 
33.72 
58,27 
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35.38 
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130'  i  3  57  57.75 


29.66 
29.79 
29.91 
30.04 
30.16 
30.29 

30.41 

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30.66 
30.78 
30.90 
3103 

3114 
3«-27 
31-39 
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31-75 

31-87 
32.00 
32.11 
32-24 
32-35 
32-47 

32-59 
32-70 

32.82 

32.94 
33-06 
33-17 

33- 29 
33-41 
33-52 
33-<>4 
33-76 
33-87 

33-98 
34- to 
34-21 
34-33 
34-44 
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34-67 
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34-89 
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35-23 

35-34 
35-45 
35-55 
35-67 
35-78 
35-89 

36,00 
36. 1 1 
36.22 
36-32 
36-44 
36-54 


1.29 

35117 
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37903 
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39840 
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41230 
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59399 
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01074 
01489 

01903 
1.29 


476 
476 

475 
474 
473 
472 

47« 
470 
468 
468 
466 
465 

465 
464 

463 
462 
461 
460 

459 
458 
456 
456 
454 
453 

453 
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450 
450 
44S 

447 

447 
445 
444 
444 
443 
441 

440 
439 
43^ 
437 
435 
434 

434 
432 
43' 
430 
429 
42S 

427 
426 

425 
424 
423 
422 

421 
419 
418 
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415 
414 


240' 

50' 
40 
30 
20 
10 

239' 

50' 

40 

30 

20 

10 

238° 

50' 
40 
30 
20 
10 

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30 
20 
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50 
40 
30 
20 
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235° 

50' 
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30 
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10 

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50' 
40 
30 
20 
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233' 

50' 
40 
30 
20 
10 

232' 

50' 
40 
30 
20 
10 

231^ 

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40 
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20 
10 

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130' 

10' 
20 
30 
40 
50 

131° 

10' 
20 
30 
40 
50 

132^ 

10' 
20 
30 
40 
50 

133' 

10' 
20 
30 
40 
50 

134' 

10' 
20 
30 
40 
50 

135' 

10' 
20 
30 
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B 


3  57  57.75 

3  57  21.10 

3  60  44.35 

3  50  7.48 

3  55  30.61 

3  54  53.43 

3  64  10.25 

3  53  38.97 


3  53 
3  52 
3  51 
3  51 


1.58 
24.08 
40.48 

8.78 


3 
3 
3  49 


50  39.97 
49  53.00 
15.04 
3  48  30.92 
3  47  68.70 
3  47  20.38 

3  40  41.95 


3 


40  3,42 
3  45  24,79 
3  44  40.00 
3  44  7.23 
3  43  2S.30 

3  42  49.27 
3  42  10.14 
3  41  30.91 
3  40  51.58 
3  40  12.15 
3  39  32.02 


3  3S 
3  38 
3  37 
3  30 
3  30 


53.00 
13.28 
33.40 
53,54 
13,52 
35  33.41 


136' 

3 

34  53.21 

10' 

3 

34  12.90 

20 

3 

33  32,50 

30 

3 

32  52.01 

40 

3 

32  11.42 

50 

3 

31  30.73 

137' 

3 

30  49.95 

10' 

3 

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20 

3 

29  2S.11 

30 

3 

28  47.05 

40 

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28  5.90 

50 

3 

27  24.00 

138' 

10' 

20 

30 
40 
50 

139° 

10' 
20 
30 
40 

50 

140° 


3  20  4.3,32 

3  20  1.S9 

3  25  20,37 

3  21  3S,70 

3  23  57.05 

3  2.!  15.20 

3  22  33.38 
3  21  51.41 
3  21  9, .35 
3  20  27.20 
3  19  4  4.90 
3   19     2.03 

3   18  20.22 


36-65 

36.75 
36.87 

36-97 
37.08 

37-«8 
37-28 
37-39 
37-50 
37.60 

37-70 
37-81 

37-9" 
38-02 
38.12 
38.22 
38.32 
38.43 

38-53 
38-63 
38-73 
38-83 
38.93 
39-03 

39- "3 
39-23 
39-33 
39-43 
39-53 
39-62 

39.72 
39.82 

39.92 
40,02 
40.11 
40.20 

40.31 
40.40 
40.49 
40.59 
4^.69 
40.78 

40.87 
40.97 
41.06 
41.15 
41.24 
41.34 

41.43 
41.52 
41,61 
41.71 

41.79 

41.88 


Log.  r 


1.29 

61903 
62317 
62729 
63140 
63550 
63959 

64300 
04773 
05179 
65583 
65987 
06389 

60790 
07190 
67588 
07985 
08382 
68777 

69171 
69504 
69955 
70345 
70734 
71122 

71509 
71895 
72270 
72602 
73044 
73425 

73805 
74184 
74501 
74937 
75312 
75080 

70059 
70430 
7  0800 
77109 
77537 
77903 

78208 
7S032 
7S995 
79350 
79710 
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80433 

80789 
81144 
81498 
81S51 
82202 

82552 


414 

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411 
410 
409 
407 

407 
406 
404 
404 
402 
401 

400 
398 

397 
397 
395 
394 

393 
391 
390 
389 

388 

387 
3S6 
384 
383 
382 
381 
380 

379 
377 
376 
375 
374 
373 

37» 
370 
369 
368 
366 

365 

364 
363 
361 
360 
359 
358 

356 
355 
354 
353 
35  > 
350 


41.07 

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349 

42,06 

83249 

348 

42.15 

83595 

346 

42.24 

83910 

345 

42.33 

84284 

344 

42.41 

343 

84027 
1.29 


230' 

50' 
40 
30 
20 
10 

229' 

50' 
40 
30 
20 
10 

228° 

50' 

40 

30 

20 

10 

227" 

50' 
40 
30 
20 
10 

226' 

60' 
40 
30 
20 
10 

225' 

50' 
40 
30 
20 
10 

224° 

60' 
40 
30 
20 
10 

223° 

50' 
40 
30 
20 
10 

222° 

50 
40 
30 
20 
10 

221° 

50' 
40 
30 
20 
10 

220' 


I  i 


!  \ 


IIP 


222 


THK    ORBIT   OF   URANUS. 


TAUl 
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9 

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220' 

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2  33  20.93 

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260 

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3  17  37.72 

42-5° 

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341 

50' 

10' 

2  32  39.79 

47.14 

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20 

3  l(i  55.13 

42-59 

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340 

40 

20 

2  31   52.59 

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30 

3  l(i   12.45 

42.68 

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339 

30 

30 

2  31     5.32 

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259 
258 
256 

30 

40 

3  15  2!l.(18 

42-77 

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20 

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2  30  17.98 

47  34 

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20 

50 

3  14  4U.83 

42-85 

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10 

60 

2  29  30.58 

47-40 

04093 

10 

42-94 

335 

47-47 

25s 

141° 

3  14     3.89 

80050 

219° 

151° 

2  28  43.  n 

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209° 

10' 

3  13  20.87 

43.02 

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50' 

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2  27  55.58 

47-53 
47.60 
47.66 

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20 

3  12  37.78 

43- « I 

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332 

40 

20 

2  27     7.98 

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30 

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30 

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2  20  20,32 

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251 

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40 

3  11   11.28 

43.28 
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330 

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2  25  32.59 

47-73 

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20 

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3  10  27.92 

88312 

329 

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50 

2  24  44.80 

47-79 
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249 

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327 

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3     9  44.47 

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48.04 

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3     3     9.09 

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313 
311 

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48.58 

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231 

230 

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144° 

3     0  5n.r,5 

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154° 

2  14  17.95 

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10' 

3     0   12.14 

44.51 

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310 

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2   13   29.31 

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2  59  27.55 

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309 

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48. 69 

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227 
226 

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224 

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2  57   13.32 

44.82 

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305 

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50 

2  10  14.18 

09831 

223 

10 

44.90 

304 

48.92 

222 

145° 

2  50  28.42 

44-98 

94.109 

215' 

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2     9  25,20 

10053 

220 
219 

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2  55  43.44 

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2      8   30.28 

48.98 

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2  54  5S.39 

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94  913 

301 

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2     7  47.24 

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216 
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2  52  42.77 

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297 

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2     5  19.81 

49.20 

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296 

49.26 

213 

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2  51  57.41 

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214° 

156° 

2     4  30.55 

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10' 

2  51    11.97 

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209 
208 
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2  48  55.21 

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2  48     9.48 

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2  47  23.07 
2  4(1  37.79 
2  45  51.83 

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46. 1 7 
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98130 
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50' 
40 

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10' 

20 

1   59  33.91 
1   58  44.29 
1   57  54. (;l 

49.62 
49.68 

12002 
12s  05 
13007 

203 
202 
200 
198 

»97 
196 

203° 

50' 
40 

30 
40 

2  45     5.80 
2  44   19.70 

98705 
9.S9SS 

284 

2*^3 

30 
20 

30 
40 

1    57      4.S8 
1   50   15.10 

49-73 
49- 78 

132(17 
13405 

30 
20 

50 

2  43  33.53 

99209 

281 

2S0 

10 

50 

1   55  25.28 

49.^2 
49.88 

13002 

10 

148° 

2  42  47.29 

46.31 
46.39 
46-45 
46.53 
46.60 
46.66 

99549 

212' 

158° 

1   54  35.40 

13798 

202° 

10' 

2  42     0.98 

99.*<27 

2  78 

50' 

10' 

1   53  45.4  7 

49-93 

13992 

194 

60' 

20 

2  41    14.59 

*(I01()4 

277 

40 

20 

1   52  65.50 

49-97 

14185 

193 

40 

30 

2  40  28.  U 

*(in3S0 

276 

1     30 

30 

1   52     5.48 

50-02 

14377 

192 

30 

40 

2  39   41.01 

*(l(m54 

274 

20 

40 

1   51    15.41 

50.07 

14507 

190 
189 
188 

20 

50 

2  38  55.01 

♦00920 

272 
271 

10 

50 

1   50  25.29 

50.12 
50- >  7 

14750 

10 

149' 

2  38     8.85 

46. 73 

♦01197 

i  211' 

159-= 

1    49  .35.12 

14944 

1S6 
185 
•83 
182 
180 

201° 

10' 

1  2  37   21.02 

♦014  07 

270 

;  50' 

10' 

1    48  44,91 

50-21 

15130 

50' 

20 

2  30   34.81 

46.  Si 
46.87 

♦O1730 

26() 

40 

20 

1    47   54,05 

50.26 

15315 

40 

80 

2  35   47.94 

♦02004 

26S 

30 

30 

1    47     4,34 

50-31 

15498 

30 

40 

2  35      I. 01 

46.93 

♦02270 

266 

!     20 

40 

1    40   13.98 

50.36 

15080 

20 

50 

{  2  34   14.00 

47.01 

♦02535 

265 

!       10 

50 

1    45  23.58 

50.40 

15800 

10 

47.07 

263 

5...  44 

•79 

160' 

2  33   20.93 

♦0279^^ 

210' 

160 

1    44   33.14 

10039 

200° 

1      „ 

1 

♦1.30 

'<      !/ 

1 

ISO 

__ 

THE   Oil  11  IT   OF   URANUS. 


223 


TABLK   VII 

.—  Cont 

inited. 

9 

E 

Log.  r 

0 

K 

Log. 

r       1 

Q         1             II 

>t 

l.SO 

0       1          n 

II 

1.80 

IQO^ 

1  44  33.14 

10039 

200° 

170° 

0  52  57.33 

24087 

89 
88 
86 
84 

83 
82 

190° 

10' 

1   43  42.05 

SO'49 

10210     ' 

77 
76 

50' 

10' 

0  52     4.81 

52-5' 

24176 

50' 

20 

1   42  52.11 

50-54 

S°-5S 
50.62 
50.O7 
50-71 

10392     ' 

40 

20 

0  51   12.28 

52-53 
52-56 

52.58 
52.60 
52.62 

24204 

40 

40 

1   43     1.53 
1   41   lO.'Jl 

10500     ' 
10739     ' 

74 
73 

30 
20 

30 
40 

0  50  19.72 
0  49  27.14 

24350 
24434 

30 
20 

50 

1   40  20.24 

1C310     ; 

71 
70 

10 

50 

0  48  34.54 

24517 

10 

161° 

1  39  29.53 

17080 

68 

67 
66 

64 
63 
62 

199° 

mr 

0  47  41.92 

52.64 
52.66 
52.6.;j 
52-70 
52.72 
52-74 

24599 

80 

189° 

10' 
20 
30 

1   3,S  38.7S 
1   37  47.98 
1   3G  57.14 

5°-75 
50.80 
50.84 
50.89 

17248     ' 

17415 

17581 

50' 
40 
30 

10' 
20 

30 

0  40  49.28 
0  45  50.02 
0  45     3.93 

24079 
24758 
24835 

79 
77 
75 
74 
73 

50' 
40 
30 

40 

1  3G     0.25 

17745 

20 

in 

0  44  11.23 

24910 

20 

50 

1  35  15.33 

50.92 
50.96 

17908     J 

10 

50 

0  43  18.51 

24984 

10 

162° 

1   34  24.37 

18070 

60 

198° 

172° 

0  42  25.77 

52.76 

26057 

188° 

10' 

1   33  33.37 

51.00 

18230     ' 

50' 

10' 

0  41  33.01 

25128 

71 

50' 

20 

1  32  42.33 

5'  04 

l.')388     ' 

S^ 

40 

20 

0  40  40.24 

52-77 

26198 

70 
69 
67 
65 
64 

40 

ao 

40 

I   31  61.24 
1  31     0.12 

510') 
51.12 
51.16 
51.20 

18545     ' 
18700 

57 

55 

30 
20 

30 
40 

0  39  47.45 
0  38  64.04 

52-79 
52.81 

_  _    y  _ 

25207 
25334 

30 
20 

50 

1  30     8.90 

18854 

54 
53 

10 

50 

0  38     1.82 

52.82 
52.84 

25399 

10 

163° 

10' 

1  29  17.70 
1   28  20.52 

51-24 

19007 
19158 

51 

197° 

50' 

173° 

10' 

0  37     8.98 
0  30  10.12 

52.86 
52-87 
52.89 

26403 
25525 

62 
61 

187° 

50' 

20 

1   27  35.25 

51-27 

19308 

50 
48 

40 

20 

0  35  23.25 

25580 

40 

30 

1   20  43.93 

51-32 

19450 

30 

30 

0  34  30.30 

25045 

59 
58 
56 

30 

40 

1   25  52.58 

5«-35 

19003 

47 
46 

20 

40 

0  33  37.45 

52.91 

25703 

20 

60 

1   25     1.19 

5'-39 

19749 

10 

50 

0  32  44.53 

52.92 

25759 

10 

S»-42 

44 

52-93 

55 

164° 

1   21     9.77 

51-46 

19893 

193° 

174° 

0  31   51.00 

25814 

53 
52 
50 
48 

47 
46 

186' 

10' 

1   23  1S.31 

20035 

42 

50' 

10' 

0  30  58.00 

52-94 
52.96 

25807 

50' 

20 

;   22  20.81 

5'-5o 

20170 

41 

40 

20 

0  30     5.70 

25919 

40 

30 
40 

1   21  35.28 
1   20  43.71 

51-53 
5>-57 
51.60 

51-64 

20310 
20454 

40 
138 
■•^6 
'35 

30 
20 

30 
40 

0  29  12.73 
0  28  10.75 

52.97 
52.98 

25909 
20017 

30 
20 

50 

1    19  52.11 

20590 

10 

50 

0  27  20.70 

52-99 
53-00 

2(!004 

10 

165° 

10' 
20 

1    19     0.17 
1    18      8.80 
1    17  17.10 

5'-67 
5'-7° 

20725 
20858 
20090 

32 

195' 

50' 
40 

175° 

10' 
20 

0  20  33.70 
0  25  40.75 
0  24  47.72 

53-01 

53-03 
53-04 
53-04 
53-o6 
53-07 

20110 
20154 
20197 

44 
43 
42 
40 
39 
37 

185° 

50' 
40 

30 

1    10  25.30 

5'-74 

21121 

31 

30 

30 

0  23  54.08 

20239 

30 

40 

1    15  33.59 

5^-77 
51. So 

5'-83 

21250 

29 

20 

40 

0  23     1.04 

20279 

20 

50 

1    14  41.79 

21377 

27 
26 

10 

50 

0  22     8.58 

20318 

10 

166° 

1    13  40.90 

5 1. 86 

21503 

194° 

176° 

0  21  15.51 

S3-08 
53.08 

20355 

35 

184° 

10' 

1    12  58.10 

21028 

'25 

50' 

10' 

0  20  22.13 

20300 

50' 

20 

1    12     0.20 

51.90 

21751 

'23 

40 

20 

0  10  29.35 

20424 

34 

40 

30 

1    11    14.27 

51-93 
51.96 
51.98 
52.02 

21873 

22 

30 

30 

0  18  30.25 

53-'o 
53-'o 
53-12 
53-12 

20450 

32 

30 

40 

1    10  22.31 

21993 

20 

20 

40 

0   17  43.15 

20480 

30 

20 

50 

1     9  30.33 

22112 

'9 
118 

10 

50 

0  10  50.03 

20515 

29 

28 

10 

167° 

10' 

1     8  38.31 

1      7   40.27 

52-04 
52.08 

22230 
22;!40 

ri6 

193° 

50' 

177° 

10' 

0   15  50.01 
0  15     3.78 

53-13 
53  13 
53.14 
53- '5 
53-15 
53-15 

20543 

20509 

26 

2  z 

183° 

50' 

20 

1      0  54.19 

22400 

114 

40 

20 

0   14  10.05 

20504 

23 

23 

40 

30 

1     0     2.08 

52.11 

22573 

'■3 

30 

30 

0  13  17.51 

20017 

30 

40 

1     5     9.94 

5-'-' 4 

52.16 

52-19 

22084 

( 1 1 

20 

40 

0   12  24.30 

20039 

20 

20 

50 

1     4  17.78 

22793 

109 
108 

10 

50 

0  11  31.21 

2G059 

19 

10 

163° 

1     3  25.59 

22901 

192' 

178' 

0  10  38.00 

53- J  6 
53.16 
5316 

20078 

1 7 

182° 

10' 

1      2   33.37 

52.22 

23008 

107 

50' 

10' 

0     0  44.00 

20005 

16 

60' 

20 

1     1  41.13 

5  =-24 

23113 

■05 

40 

20 

0     8  51.74 

20711 

■5 

13 
1 J 

40 

30 

1      0   48.80 

52.27 

2.3217 

104 

30 

30 

0     7  68.58 

20720 

30 

40 

0  59  50.57 

52-29 

23320 

'O,?  _ 

20 

40 

0     7     5.41 

53-17 

20739 

20 

50 

0  59     4.25 

52-32 
52.35 

23421 

lol 

100 

10 

50 

0     0  12.24 

53-17 
53-17 

20750 

10 

10 

169° 

0  58  ii.no 

23521 

191° 

173° 

0     5  10.07 

53-18 

20700 

8 

181° 

10' 

0  57   19.53 

52-37 

23019 

98 

50' 

10' 

0     4  25.89 

20708 

50' 

20 

0  50  27.13 

52-40 

23710 

97 

40 

20 

0     3  32.72 

53-17 

53-18 
53-18 
53-18 

20775 

7 

40 

30 

,  0  55  34.71 

52-42 

23811 

95 

30 

30 

0     2  39.54 

20780 

5 

30 

40 

'  0  54  42.27 

5244 
52-46 

23904 

93 

20 

40 

0     1   40.30 

20784 

4 

20 

50 

0  53  49.81 

23990 

92 

10 

60 

0     0  53.18 

20780 

2 

10 

52.49 

91 

53-18 

1 

170' 

0  52  57.32 

1.30 

190° 

180° 

0     0     0.00 

20787 
1.30 

180° 

<J 

'  ''1 

J 

i  9 

1 

If 

■  iii  ■] 

4't 

i|:  :ii 


i 


I!     \\ 


l.cHi 


m 


11 


224 


TllK    OllBlT   OF   UUANUS. 


TAISLK  VIII,  Ana. 

1. — Action  of  Juimter. 

Arjt. 

(c.f.O)        Dili'. 

(,-.,..  1) 

It 

0.14 

(u.c.l) 

(U.S.  2) 

{v.cM) 

(p.c.O) 
2331 

(p.«.l) 

1  (p.f.l) 

0 

55.03  ,' 
55.58+°- 55 

It 

4.42 

0.10 

tl 

0.11 

104 

1 

134 

1 

0.13 

4.42 

0.10 

0.12 

2.-!31 

100 

134 

2 

5(i.l3   °-5S 
5(;.(I8  °-55 

.,...   Ill 

59.97   °55 
0-54 

«i.o(i+°-ss 

Cl.dO   °-S4 

C2.15   °-5S 

(;2.(!9   °-54 

0-54 

0.12 

4.41 

0.10 

0.12 

2330 

107 

134 

3 

0.11 

4.41 

0.10 

0.12 

2330 

109 

134 

4 

O.ll 

4.41 

0.10 

0.12 

2330 

171 

135 

5 

0.10 

4.40 

0.10 

0.13 

2329 

172 

135 

C. 

0.09 

4.40 

o.ll 

0.13 

2328 

174 

135 

■: 

0.09 

4.39 

0.11 

0.13 

2328 

170 

135 

8 

0.08 

4.39 

0.11 

0.13 

2327 

177 

135 

9 

0.08 

4.38 

0.11 

0.14 

2320 

179 

130 

10 

0.08 

4.38 

0.11 

0.14 

2324 

180 

130 

11 

0.07 

4,37 

0.11 

0.14 

2323 

182 

130 

12 

0.07 

4.37 

0.11 

0.15 

2321 

184 

137 

13 

0.07 

4.30 

0.11 

0.15 

2320 

185 

137 

U 

0.08 

4.30 

0.11 

0.15 

2318 

187 

137 

15 

03.23 

0.08 

4.35 

0.11 

0.15 

2310 

1S8 

137 

l(i 

03.78 +°-SS 

0.08 

4.35 

0.11 

0.10 

2311 

190 

137 

17 

04.32   °-54 

0.09 

4.34 

0  11 

0.10 

2312 

191 

138 

IS 

04.80   °-54 

0.09 

4.34 

0.11 

0.10 

2310 

193 

138 

■J 

05.40   °S4 
0-53 
05.93 

0.10 

4.33 

0.11 

0.17 

2308 

194 

138 

20 

0.11 

4.33 

o.ll 

0.17 

2305 

190 

138 

21 

00.47 +°S4 

0.12 

4.32 

o.ll 

0.17 

2303 

198 

139 

22 

07.00   °-53 

0.13 

4.32 

0.11 

0.17 

2300 

199 

139 

23 

07.54   °-54 

0.14 

4.31 

0.11 

0.18 

2297 

200 

139 

24 

08.07   °-53 
0-53 
08.00  , 

0. 15 

4.30 

0.11 

0.18 

2294 

202 

140 

25 

0.10 

4.30 

0.10 

0.18 

2291 

203 

140 

2(! 

09.14 +°S4 

0.18 

4.29 

0.10 

0.19 

2288 

205 

140 

27 

09.06   °-52 

0.20 

4.29 

0.10 

0,19 

2284 

200 

140 

2S 

70.19   °-53 

0.21 

4.28 

0.10 

0,19 

2281 

208 

140 

29 

70.72   -53 

71.24  , 
71.70 +°-5^ 

0.23 

4.28 

0.10 

0.20 

2277 

209 

141 

30 

0.25 

4.27 

0.10 

0.20 

2274 

210 

141 

31 

0.27 

4.27 

0.10 

0.20 

2:i70 

212 

141 

32 

7->  28   °'5^ 

0.29 

4.20 

0.10 

0  21 

2200 

213 

141 

33 

72^80   °-5^ 

0.31 

4.20 

0.10 

0.21 

2202 

214 

142 

34 

73.32  III 
74.34 +°- 

0.33 

4.25 

0.10 

0.21 

2258 

210 

142 

35 

0.35 

4.25 

0.10 

0.22 

2254 

217 

142 

30 

0.38 

4.24 

0.10 

0.22 

2249 

218 

142 

37 

74.85   °- 

0.40 

4.24 

0.10 

0.22 

2245 

219 

143 

38 

75.30   °-5 

0.43 

4.23 

0.10 

0.23 

2240 

221 

143 

39 

0.30 
70.87 +°--2° 

0.40 

4.23 

0.09 

0.23 

2230 

222 

143 

40 

0.49 

4.22 

0.09 

0.23 

2231 

223 

143 

41 

0.52 

4.22 

0,09 

0,23 

2220 

224 

144 

42 

77.37   °-i:° 

0.55 

4.22 

0,09 

0,24 

2221 

225 

144 

43 

78.80 

0.58 

4.21 

0.09 

0,24 

2210 

227 

144 

44 

0.01 

4.21 

0.09 

0,24 

2210 

228 

144 

45 

0.04 

4.21 

0.09 

0,25 

2205 

229 

144 

4(i 

79.35+°- 49 

0.08 

4.20 

0.09 

0.25 

2200 

230 

145 

47 

79.84   °-49 
80.32   °-4f 
80.80   °-48 
0.48 
81.28  , 
81.70 +°-48 

0.71 

4.20 

0.09 

0.25 

2194 

231 

145 

48 

0.75 

4.20 

0.09 

0.20 

2188 

232 

145 

49 

0.79 

4.20 

(.09 

0.20 

2182 

233 

145 

50 

0.82 

4.20 

0.08 

0.20 

2170 

234 

140 

51 

0,80 

4.19 

0.08  . 

0.20 

2170 

235 

140 

52 

82.23   °-47 

0.90 

4.19 

0.08 

0.20 

2104 

230 

140 

53 

82,70   °-*7 

0.94 

4.19 

0.08 

0.27 

2158 

237 

140 

54 

83.17   °-+J 
0.46 

0.98 

4.19 

0.08 

0.27 

2151 

238 

140 

55 

83.03 

1.02 

4.19 

0.07 

0.27 

2145 

239 

140 

5li 

84.10 +°-*7 

1,07 

4.19 

0.07 

0.28 

2138 

240 

140 

57 

84.55   °-45 
85.01  °;f 

1.11 

4,19 

0.07 

0.28 

2132 

241 

140 

5S 

l.lfi 

4,19 

0.07 

0.28 

2125 

242 

147 

59 

85.40   °-*5 
0-4.1 
85,91 

1.20 

4.19 

0.07 

0.28 

2118 

243 

147 

r.o 

1.25 

4.19 

0.07 

0.29 

2111 

241 

147 

TUE   OllBIT 

OF   URANUS 

225 

TA15LE 

VllI,  Ano.   \.—C 

inlinii.d. 

Arg. 

no 

(ii.f.O)       Diff. 

(o.s.l) 

(v.c.X) 

(».«.2) 

It 

0.07 

(i'.f.2) 

(p.r.O) 

(,,.«.  1)  1 

(p.O'.l) 

II         II 
85.01  . 

1.25 

4.10 

II 
0.29     j 

2111 

244 

147 

(ii 

80.30 +°-'*5 

1.30 

4.19 

0.07     1 

0.20 

2104 

244 

147 

(12 

80.80     °-*^ 

1  34 

4.20     1 

0.00 

0.29 

2007 

245     I 

148 

03 

87.24     °-*-* 

1.39 

4.20 

0.00 

0.29 

2000 

240 

148 

04 

8  .08     °-*-* 

1.44 

4.20 

0.00 

0.29 

2082 

247 

148 

05 

0.43 
8S.11  . 

1.49 

4.20 

con 

0.30     I 

2075 

248 

148 

Oli 

88.53+°-*' 

1.54 

4.21 

0.00 

0.30 

2007 

248 

MS 

07 

88.00     °-« 

1.00 

4.21 

0.00 

0,30 

2059 

249 

148 

OS 

80.38     °--*' 

1.05 

4.21 

0.05 

0.30     ' 

2052 

250 

148 

0!) 

80.80   °;]J 

90.21  , 

1.70 

4.22 

0.05 

0.31 

2044 

250     1 

148 

70 

1.75 

4.22 

0.05 

0,31 

2030 

251 

149 

71 

00.02+°- 4' 

1.81 

4.23 

0.05 

0.31      i 

2028 

253 

149 

7i 

91.03     "-4' 

1.80 

4.23 

0.05 

0.31  ; 

2020 

252 

149 

7:i 

91.43     °»° 

1.92 

4.24 

0.05 

0.31      I 

2012 

253 

149 

74 

91.83     °-+^ 

1.98 

4.24 

0.04 

0.32 

2003 

253 

149 

7.1 

0-39 
92.22  , 

2.03 

4.25 

0.04 

0.32 

1995 

254 

149 

7."> 

02. 02+°- 4^ 

2.09 

4.20 

0.04 

0.32 

1980 

254 

149 

77 

ot.oo   °-^i 

2.15 

4.20 

0.04 

0.32 

1978 

255 

149 

7'f 

9.!.3S     °--^f 

2.21 

4.27 

0.04 

0.32 

1000 

255 

149 

7'.» 
80 

93.7li     °--]^ 
0.38 

94.14  , 

2.27 
2.33 

4.2s 
4.29 

0.04 
0.04 

0.32 
0,32 

1000 
10.--)2 

25  G 
250 

149 
149 

81 

94.51 +°-.>7 

2.30 

4.30 

0.03 

0,32 

r.M3 

257 

149 

82 

04.87     °--5^ 

2.45 

4.31 

0,03 

0.33 

1934 

257 

149 

83 

95.23     °-^^ 

2.51 

4.32 

0.03 

0,33 

1925 

257 

149 

84 

95.50     °-^^ 
0.36 

05.05  , 

2.57 

4.33 

0.03 

0.33 

1010 

258 

149 

85 

2.04 

4.34 

0,03 

0.33 

1000 

258 

149 

80 

9!. 30  *-°--^5 

2.70 

4.35 

0.03 

0,33 

1807 

258 

149 

87 

9;.04    °'5-* 

2.70 

4.30 

0.03 

0.33 

1SS8 

259 

149 

8S 

01!.  OS     °--'  + 

2.  S3 

4.37 

0.03 

0.S3 

1878 

250 

149 

8!> 

07.32     -35 
97.05  , 

2.80 

4.38 

0.02 

0.33 

1809 

259 

149 

00 

2.90 

4.30 

0.02 

0..33 

1859 

200 

148 

01 

07.07 +°--'^ 

3,02 

4,41 

0.02 

0.33 

1850 

200 

148 

02 

9^.20     °-3-' 

3.00 

4,42 

0.02 

0.33 

1840 

ac'o 

148 

03 

os.oi    °-3^ 

3.15 

4,43 

0.02 

0.33 

1830 

200 

148 

94 

OS.  02     °-3' 
0.31 

90.23  , 

3.22 

4.45 

0,02 

0.33 

1820 

200 

148 

05 

3.29 

4.40 

0.02 

0.33 

ISIO 

201 

148 

01") 

90  53+°- 3° 

3,35 

4.4S 

0.02 

0,34 

l.'^OO 

201 

148 

07 

00.  S3     °-3° 

3.42 

4.40 

0.02 

0.34 

1790 

201 

148 

OS 

100.12     °"} 

3,49 

4.51 

0,01 

0.34 

1780 

201 

m 

00 

100.70  , 

3.50 

4.52 

0.01 

0.34 

1770 

201 

147 

100 

3.03 

4.54 

0.01 

0.34 

1700 

201 

147 

101 

100.97+°-; 
101.25   ° :, 

3.00 

4.50 

0.01 

0.34 

1750 

201 

147 

102 

3,70 

4,57 

0.01 

0.34 

1739 

201 

147 

103 

•"'.'V2   : 

3.83 

4,50 

0.01 

0.34 

1729 

201 

140 

104 

3.00 

4. .01 

0.01 

0.34 

1718 

201 

140 

1 05 

102.04  , 

3,97 

4,03 

0,01 

0,34 

:     1 708 

201 

140 

100 

102.29+°-   S 

4,04 

4,05 

0.01 

0,34 

i      1097 

201 

145 

107 

102.54     °-',^ 

4,11 

4.07 

0.01 

0.34 

lOSO 

201 

145 

lOS 

102. 7S     °-   1 

4.18 

4.09 

0.0 1 

0,34 

1070 

201 

145 

109 

103.02   °;;J 

103.25 

4.25 

4.71 

0.01 

0,34 

1005 

201 

145 

110 

4.32 

4.73 

1      0.01 

0.34 

1054 

201 

144 

111 

103.48+°- '5 

4.30 

4.75 

0,01 

0.34 

1043 

201 

144 

112 

103.70     °-" 

4.40 

4.77 

0.01 

0.34 

1032 

201 

144 

113 

103.02     °-" 

4,53 

4,70 

0,01 

0.34 

1022 

200 

143 

114 

10.13  -; 

101.34  , 

4.00 

1      4.81 

0,01 

0.34 

1011 

200 

143 

115 

4.07 

4,83 

0.01 

0.33 

1000 

200 

142 

lift 

104.54+°-^° 

4.74 

4.S5 

0,01 

0,33 

15SS 

200 

142 

117 

104.74     °-"° 

4.81 

4.87 

0,01 

0.33 

1577 

200 

142 

118 

104.03     °-'^ 
l«^ll     0.8 

4.88 

4.90 

O.Ol 

0.33 

1500 

259 

141 

119 

4.05 

4.92 

0.01 

0.33 

1555 

250 

141 

120 

105.29 

5.01 

4.94 

0,02 

0.33 

1544 

250 

140 

1 

19      July,  1878. 

„!     \ 


if! 


!«i    I 


■  I 


226 


THE   ORBIT   OF   UIIANUS. 


TAIILF 

VIII,  A 

no.  1. — Continued. 

Ai'n. 

(iJ.c.O)   Diir. 

(i>.».l) 

{v.cA) 

(V.H.i) 

(0.C.2) 

(p.f.O) 

(p.».l) 

(P.f.l) 

t,             n 

n 

tt 

tt 

tt 

120 

•"■'■'•2«  ,„  ., 

5.01 

4.94 

0.02 

0.33 

1544 

259 

140 

121 

105.40+°- 

5.08 

4.97 

0.02 

0.33 

1533 

259 

140 

122 

105.03  °-'/ 

105.79  °'^ 

105.95  °"^ 
o>  1  ^ 

6.15 

4.99 

0.02 

0.33 

1521 

258 

139 

12;j 

5.22 

5.01 

0.02 

0.33 

1510 

258 

1 39 

124 

6.29 

6.04 

0.02 

0.33 

1499 

258 

138 

135 

100.10  , 
100.25+°- ;S 

5.30 

6.00 

0.02 

0.33 

1487 

257 

138 

12(! 

6.43 

6.09 

0.02 

0.33 

1470 

257 

137 

127 

100.39  °-^^ 

6.50 

6.11 

0.02 

0.33 

1404 

257 

137 

12S 

100.53  °-  j 

6.57 

5.14 

0.02 

0.33 

1453 

250 

130 

129 

100.05  °;;; 

100.78 

6.04 

6.16 

0.02 

0.33 

1441 

250 

135 

IM 

5.71 

6.19 

0.02 

0.32 

1430 

250 

135 

l.Jl 

100.«9+°-" 

6.78 

5.21 

0.03 

0.32 

1418 

255 

134 

l;!2 

107.01  °-'* 

6.85 

6.24 

0.03 

0.32 

1407 

255 

134 

l:!.l 

107.11  °-'° 

5.91 

6.20 

0.03 

0.32 

1395 

254 

133 

131 

107.22  °-" 
0.09 

6.98 

5.29 

0.03 

0.32 

1384 

254 

132 

135 

107.31  , 

fi.05 

6.32 

0.03 

0.32 

1373 

254 

132 

13(! 

107.40+°-°'> 

fi.lt 

6.34 

0.04 

0.32 

I.'IOO 

253 

131 

137 

107.49  °°'l 
107.57  °-°'^ 

fi.18 

6.37 

0.04 

0.32 

1349 

253 

131 

13S 

0.25 

6.40 

0.04 

0.32 

1337 

252 

130 

139 

107.04  °-°/ 
0.06 

C.31 

6.42 

0.04 

0.32 

1325 

252 

129 

140 

107.70 

107.70+°°^ 
107. S2  °-°'^ 

0.38 

6.45 

0.04 

0.32 

1314 

251 

129 

141 

fi.44 

6.48 

0.05 

0.31 

1 302 

251 

128 

142 

G.51 

5.51 

0.05 

0.31 

1290 

250 

127 

143 

107..S7  °-°5 

0.57 

6.53 

0.05 

0.31 

1278 

250 

120 

144 

107.91  °-°-t 
0.04 

107.95 

fi.G4 

6.50 

0.05 

0.31 

1207 

249 

120 

1!5 

0.70 

5.59 

(  5 

0.31 

1 255 

249 

125 

lie 

107.98 +°-°3 

0.70 

6.01 

0.06 

0.31 

1243 

248 

124 

147 

los.oi  °-°-< 

0.83 

5.04 

0.00 

0.31 

1231 

248 

123 

US 

10X.03  °-°^ 

fi.H9 

5  07 

0,00 

0.31 

1220 

247 

123 

149 

108.05  °-°^ 

0.00 

0.95 

5.09 

0.00 

0.31 

1208 

247 

122 

150 

108.05  , 

7.01 

5.72 

0.07 

0.31 

1190 

240 

121 

ir.i 

108.00+°°' 

7.08 

6.75 

0.(»7 

0.31 

11S4 

240 

120 

152 

108.05-°°' 

7.14 

5.78 

0.07 

0.31 

1172 

245 

119 

153 

108.04  °-°' 

7.20 

5.  SO 

0.07 

0.30 

1101 

244 

118 

154 

108.03   °-°' 
0.02 

7.25 

6.83 

O.OS 

0.30 

1149 

244 

118 

155 

i-^.Ol 

7.31 

6.80 

0.08 

0.30 

1137 

243 

117 

15r, 

,  'T.98-°-°-' 

7.37 

6.S8 

0.08 

0.:i0 

1 1  20 

243 

110 

157 

107.95  °-°-] 

7.43 

5.91 

0.09 

0.30 

III4 

242 

115 

15S 

'"'•"»  TA 

7.49 

6.94 

0.09 

0.30 

1102 

241 

114 

159 

107.87  °-°! 
0.05 

107.77  °-°S 

7.54 

6.97 

0.09 

0.30 

1090 

241 

113 

IfiO 

7.00 

6.99 

0.10 

0.30 

1079 

240 

112 

1(11 

7.00 

6.02 

0.10 

0,30 

1007 

9  to 

112 

1(;2 

107.70  °-°/ 
107.04  °-°^ 
107.56  °-°^^ 

0.07 

107.49 

7.71 

0.04 

0.10 

0.30 

1055 

239 

111 

103 

7.77 

0.07 

0.11 

0,30 

1044 

23S 

110 

1G4 

7.82 

6.10 

O.ll 

0.29 

1032 

238 

109 

ir,5 

7.87 

0.12 

0.11 

0.29 

1020 

237 

108 

ii'.a 

7.93 

6.15 

0.12 

0.29 

1009 

230 

107 

107 

107.31  °-°^ 
107.22  °-°? 

7.98 

6.17 

0.12 

0.29 

997 

230 

100 

lOS 

8.03 

6.20 

0.12 

0.29 

!)S0 

235 

105 

109 

8.08 

6.22 

0.13 

0.29 

974 

234 

104 

170 

107.01 

8.13 

6.25 

0.13 

0.29 

903 

234 

103 

171 

100.89-°-'^ 

8.18 

6.27 

0.14 

0. 29 

951 

2.-13 

102 

172 

100.77  °-'-' 

8.23 

6.29 

0.14 

0.29 

940 

232 

101 

173 

100.05  °-" 

8.28 

6.32 

0.14 

0.29 

928 

232 

100 

174 

100.52  °  '•' 
0.14 

100.38 

8.33 

6.34 

0.15 

0.29 

917 

231 

99 

175 

8.37 

6.30 

0.15 

0,29 

900 

230 

98 

170 

100.24-°-'-* 

8.42 

6.39 

0.15 

0.29 

894 

230 

97 

177 

100.10  ° ';} 

105.94  °-'^ 

105.78  °-'^ 
0. 16 

8.40 

6.41 

0.10 

0,29 

8s:i 

229 

90 

17S 

8.51 

6.43 

0.10 

0. 29 

872 

228 

95 

179 

8.55 

6.45 

0.17 

0.29 

801 

228 

94 

180 

105.02 

8,00 

6.47 

0.17 

0.29 

850 

227 

93 

THE   OIllJIT  OP   URANUS. 


227 


TAlUiK 

VIII,  Ann.  \.—C 

onlinued. 

Arg. 

(u.c.O)   Diir 

(1..8.I) 

{>y.v.\) 

(.'.«.2) 

(y.f.2) 

(f.c.O) 

(p.O.l) 

(p.C.l) 

180 

105.f.2 

8.00 

n 

6.47 

0.17 

0.29 

850 

227 

93 

181 

105.45-°-  7 
105.27  °'  ^ 
105.09  °-  '^ 

8.04 

6.50 

0.17 

0.29 

839 

220 

92 

182 

8.08 

0.52 

0.18 

0.29 

828 

220 

91 

183 

8.72 

0.54 

0.18 

0.29 

810 

225 

90 

184 

104.90  °;;j 

8.7(1 

C.50 

0.19 

0.29 

805 

224 

89 

185 

8.80 

fi.57 

0.19 

0.29 

794 

224 

88 

18(> 

104.51-°'^° 

8.84 

C.59 

0.19 

0,29 

784 

223 

87 

ls7 

104.31  °-\° 

8.88 

6.01 

0.20 

0.29 

773 

222 

80 

188 

104.10  °- 

8.92 

0.03 

0.20 

0.29 

702 

222 

85 

18« 

103.89  °:^; 

8.9(i 

0,05 

0.21 

0.29 

751 

221 

84 

190 

103.07  „  ,, 

8.99 

6.07 

0.21 

0.29 

740 

221 

83 

191 

103.44-°-^ 

9.03 

0.08 

..21 

0.29 

730 

220 

82 

192 

103.21  °-\\ 

9.07 

0.70 

0.22 

0.29 

719 

219 

81 

193 

102.98  °':'; 

9.10 

0.71 

0.22 

0.29 

708 

219 

80 

194 

102.49  „  ,. 

9.13 

0.73 

0.22 

0.29 

698 

218 

79 

195 

9.1 7 

0.74 

0.23 

0.29 

688 

217 

78 

I9f> 

102.24 -°-^S 
101.72  °,, 

9.20 

0.70 

0.23 

0.29 

677 

210 

77 

197 

9.23 

0.77 

0.24 

0.29 

607 

210 

70 

198 

9.20 

0.78 

0.24 

0.29 

057 

215 

75 

199 

101.18    , 

9.29 

6,79 

0.24 

0.29 

640 

215 

74 

200 

9.32 

6.81 

0.25 

0.29 

030 

214 

73 

201 

100.90-°-^„ 

9.35 

0.82 

0.25 

0.29 

620 

213 

72 

202 

100.(12  °-^'* 

9.38 

6.83 

0.25 

0.29 

610 

213 

71 

ms 

100.33  °'9 

9.41 

0.84 

0.20 

0.29 

600 

212 

70 

204 

100.04  °-"9 

O.JO 

9.43 

6.85 

0.20 

0.29 

590 

211 

69 

205 

99.74 

9.40 

0.80 

0.26 

0.29 

680 

211 

08 

20li 

99.44 -°-3° 

9.48 

0.80 

0.27 

0.29 

577 

210 

07 

207 

99.13  °-5' 

9.51 

0.87 

0.27 

0.29 

507 

210 

00 

208 

98.82  °--5' 

9.53 

6.88 

0.27 

0.29 

557 

209 

05 

209 

98.5(-  °-^- 
0.32 

98.18 

9.55 

0.88 

0.28 

0.29 

548 

208 

64 

210 

9.58 

6.89 

0.28 

0.29 

538 

208 

63 

211 

97. 85-°' 3^ 

9.00 

0.89 

0.28 

0.29 

629 

207 

02 

212 

97.52  °-3- 

9.02 

0.90 

0.29 

0.29 

520 

200 

01 

2l;5 

97.18  °\] 

9.04 

0.90 

0.29 

0.29 

510 

200 

00 

214 

9(1.84  "-^1 
911.49 

9.00 

0.90 

0.29 

0.29 

601 

205 

59 

215 

9.0s 

0.91 

0.30 

0.29 

492 

205 

58 

21(1 

9(1.1 4 -°-3S 

9.70 

0.91 

0.30 

0.29 

483 

204 

57 

217 

95.79  °-35 
95.43  °-t 

9.71 

0.91 

0.30 

0.29 

474 

204 

50 

218 

9.73 

0.91 

0.31 

0.29 

404 

203 

55 

219 

95.0G  -3; 

9.75 

6.91 

0.31 

0.30 

450 

202 

54 

220 

9.70 

6.91 

0.31 

0.30 

448 

202 

53 

221 

94.32-°-" 
93.94  °-3-^ 
93.5(!  °-3« 

9.78 

6.90 

0,31 

0.30 

439 

201 

52 

222 

9.79 

6.90 

0.32 

0.30 

430 

201 

51 

22;] 

9.81 

0.90 

0.32 

0.30 

422 

200 

50 

224 

«^-^^  c:5 

9.82 

6.90 

0.32 

0.30 

413 

200 

49 

225 

92.78 

9.83 

6.89 

0.33 

0.30 

405 

199 

48 

22« 

92.39-°- 3^ 

9.85 

6.88 

0.33 

0.30 

398 

198 

47 

227 

91.99  °-^° 

9.80 

0.88 

0.33 

0.30 

390 

198 

40 

228 

»!-'"•»  o\i 

9.87 

0.87 

0.33 

0.30 

382 

197 

40 

229 

90.77 

9.88 

6.86 

0.34 

0.30 

374 

197 

45 

230 

9.89 

0.85 

0.34 

0.30 

305 

196 

44 

231 

90.30-°-^' 

9.90 

0.84 

0.34 

0.30 

357 

190 

43 

232 

89.94  °-*^ 

9.91 

0.83 

0.34 

0.30 

849 

195 

42 

233 

89.51  °  •♦3 

9.91 

0.82 

0.34 

0.30 

342 

194 

41 

234 

89.09  °--*^ 
0.44 
88.r,5 

9.92 

0.81 

0.35 

0.30 

334 

194 

41 

235 

9.93 

0.80 

0.35 

0.30 

327 

194 

40 

23() 

88.22 -°-*3 

9.94 

0.79 

0.35 

0.30 

320 

1 93 

89 

237 

87.78  °-4;^ 

9.94 

0.78 

0.35 

0.30 

312 

192 

88 

238 

87.34  °-44 

9.95 

0.70 

0.35 

0.30 

305 

192 

38 

239 

80.89  °-'*5 
0-45 
8(1.44 

9.95 

0.74 

0.30 

0.30 

298 

191 

37 

240 

9.90 

0.73 

0.30 

0.31 

291 

191 

36 

!  1; 


j'l 


^.1 


11 


228 


THE   ORBIT   OP   URANUS. 


TAHLE  VIII,  Alio.  l.—CoiUinm-<l.                                                    | 

Arg. 

(i'.<'.0)   Diff. 

(.-.«.!) 

(v.r.l) 
C.73 

(r.«.2) 

(u.c.2) 
It 
0.31 

(,,.r.O) 

(,,.«.!) 

O'.'-.l) 

240 

II          II 

9.96 

0.36 

291 

191 

36 

241 

85. 99-°- 45 
85.63  Ijl 

9,96 

6.71 

0.30 

0.31 

284 

190 

35 

342 

9.96 

0.09 

0.36 

0  31 

278 

190 

35 

24.'i 

85.08  °-'»5 

9.97 

6.67 

0.36 

0.31 

271 

189 

34 

244 

«^"»  lil 

9.97 

6.66 

0.37 

0.31 

265 

189 

33 

245 

9.97 

6.64 

0.37 

0.31 

268 

188 

33 

24fi 

83.08-°- 47 

9.97 

0.61 

0.37 

0.31 

252 

188 

32 

247 

83.21  y,i 

82.73  If. 

9.98 

0.59 

0.37 

0.31 

246 

1S7 

31 

24S 

9.98 

6.57 

0.37 

0.31 

240 

187 

31 

249 

9.98 

6.55 

0.37 

0.31 

234 

180 

30 

250 

81.77 

9.98 

6.53 

0.37 

0.31 

228 

186 

29 

251 

81.28-°-49 

9.98 

0.51 

0.38 

0.31 

222 

185 

29 

252 

80.80  °-4» 

9.98 

6.48 

0.38 

0.31 

210 

185 

28 

253 

80.30  °-5° 

9.97 

6.46 

0.38 

0.31 

211 

184 

28 

254 

79.81  °-49 
0.50 

9.97 

6.43 

0.38 

0.31 

205 

184 

27 

255 

9.97 

6.40 

0.38 

0.31 

200 

183 

27 

25  (i 

78. 82-°- 49 

9.97 

6.37 

0.38 

0.30 

195 

183 

26 

257 

78.31  °-5i 

9.90 

6.34 

0.38 

0.30 

190 

182 

26 

25S 

77.81  °-5° 

9.90 

6.31 

0.38 

0.30 

184 

182 

25 

250 

77.30  °Si 
0.51 
70.79 

9.90 

0.28 

0.38 

0.30 

179 

181 

25 

2f.O 

9.95 

6.25 

0.38 

0.30 

175 

181 

24 

20 1 

70. 28-°- 5' 

9.95 

6.22 

0.38 

0.30 

170 

180 

24 

202 

75.77  °-5' 

9.95 

6.19 

0.38 

0.30 

105 

180 

24 

2fi;i 

75.25  °-S» 

9.94 

6.16 

0.33 

0.30 

101 

179 

23 

204 

74.73  °-S* 
0.52 

74.21 

9.94 

6.13 

0.38 

0.30 

150 

179 

23 

205 

9.93 

6.10 

0.38 

0.30 

152 

178 

22 

200 

73.09-°-S2 

9.93 

6.06 

0.38 

0.30 

148 

178 

22 

207 

73.10  °-53 

9.92 

6.03 

0.38 

0.30 

144 

177 

22 

208 

72.04  °'52 

9.92 

5.99 

0.38 

0.30 

140 

177 

21 

209 

72.11  °-53 
0.54 

9.91 

5.96 

0.38 

0.30 

137 

no 

21 

270 

9.90 

5.92 

0.39 

0.30 

133 

170 

21 

271 

71.04 -°-53 

9.90 

5.88 

0.39 

0.30 

129 

175 

20 

272 

70.51  °-53 

9.89 

5.85 

0.39 

0.30 

120 

175 

20 

273 

09.97  °S4 

9.88 

6.81 

0.39 

0.30 

122 

174 

20 

274 

09.43  °S4 

■•■54 
03.89 

9.88 

6.77 

0.39 

0.30 

119 

174 

20 

275 

9.87 

5.73 

0.39 

0.30 

no 

174 

20 

270 

08.35-^-54 

9.80 

6.69 

0.39 

0.29 

113 

173 

19 

277 

07.81  °-S4 

9.85 

6.65 

0.39 

0.29 

110 

172 

19 

27.S 

07.20  °-5S 

9.84 

5.61 

0.39 

0.29 

107 

172 

19 

279 

Cfi.72  If 
00.17 

9.84 

5.57 

0.39 

0.29 

105 

171 

19 

280 

9.83 

6.52 

0.39 

0.29 

102 

171 

19 

281 

C5.02-°-5S 

9.82 

5.48 

0.39 

0.29 

100 

170 

19 

282 

05.07  °S5 

9.81 

5.44 

0.39 

0.29 

97 

170 

18 

283 

04.52  °-S5 

9.80 

5.40 

0.39 

0.29 

95 

169 

18 

284 

63.97  °-S5 
0.56 
03.41    ^ 

9.79 

5.36 

0.39 

0.29 

93 

109 

18 

285 

9.78 

6.31 

0.39 

0.28 

91 

108 

18 

286 

02.80— °- 55 

9.77 

6.26 

0.39 

0.28 

90 

108 

18 

287 

02.31  °-55 

9.77 

5.23 

0.39 

0.28 

88 

167 

18 

2SS 

01.75  °-56 

9.70 

5.17 

0.39 

0.28 

80 

167 

18 

289 

01.19  °S6 

°-5S 

00.04    ^ 

9.75 

6.13 

0.38 

0.28 

85 

100 

18 

290 

9.74 

5.08 

0.38 

0.28 

84 

100 

18 

291 

60.08— °- 5'' 

9.73 

6.03 

0.38 

0.27 

82 

165 

19 

292 

59.52  °-5^ 

9.72 

4.99 

0.38 

0.27 

81 

164 

19 

293 

58.90  °-5^ 

9.71 

4.94 

0.38 

0.27 

80 

164 

19 

294 

58.40  °-56 
0,56 

9.70 

4.89 

0.38 

0.27 

80 

163 

19 

295 

57.84     . 
57.28-^°-5° 
50.72  °-5» 

9,69 

4.84 

0.38 

0.27 

79 

163 

19 

290 

9.68 

4.80 

0.38 

0.27 

78 

102 

19 

297 

9.07 

4.75 

0.38 

■  0.26 

78 

102 

19 

298 

50.16  °-5° 

55.00  °'5o 
0.56 

5.V04 

9.00 

4.70 

0.38 

0.26 

77 

101 

20 

299 

9.65 

4.05 

0.38 

0.26 

77 

101 

20 

300 

9.64 

4.60 

0.38 

0.2') 

77 

100 

20 

THE   ORBIT   OP   URANUS. 


229 


TAIJLI' 

VIII,  A 

mi.  \.—t 

Continued. 

Arg. 

(i).r.O)    Diir. 

(...,.1) 

(v.c.l) 

(u.(..2) 

(u.e.2) 

(p.r.O) 

(p.".!) 

(,.<■.!) 

300 

n             II 

65.04_„  cfi 
5t.48-°-S^ 

53.92  °-S^ 
49.44 

9.04 

ft 

4.00 

0.38 

0.2c 

77 

160 

20 

3Ul 

9.03 

4.55 

0.38 

0.20 

77 

159 

20 

302 

9.02 

4.50 

0.38 

0.25 

77 

159 

20 

303 

9.01 

4.45 

0.38 

0.25 

77 

158 

21 

304 

9.00 

4.40 

0.38 

0.25 

78 

158 

21 

305 

9.59 

4.35 

0.38 

0.25 

78 

157 

?1 

300 

9.58 

4.30 

0.38 

0.25 

79 

150 

22 

307 

9.57 

4.25 

0.38 

0.24 

80 

150 

22 

30S 

9.50 

4.20 

0.38 

0.24 

81 

155 

22 

309 

9.55 

4.15 

0.38 

0.24 

82 

154 

23 

310 

9.54 

4.10 

0.38 

0.24 

83 

154 

23 

311 

48.88 -°-S6 

9.53 

4.05 

0.38 

0.23 

84 

153 

24 

312 

48.33  °S5 

9.52 

4.01 

0.38 

0.23 

85 

152 

24 

313 

47.77  °'5^ 

9.51 

3.94 

0.38 

0.23 

87 

152 

24 

3U 

47.22  "-55 
0.56 

9.50 

3.89 

0.38 

0.23 

88 

151 

25 

315 

40.00 

9.49 

3.84 

0.38 

0.23 

90 

150 

25 

31G 

40.11 -°-5S 

9.48 

3.79 

0.38 

0.22 

92 

150 

20 

317 

45.50  °-5S 

9.47 

3.74 

0.38 

0.22 

94 

149 

20 

318 

45.01  °SS 

9.40 

3.09 

0.38 

0.22 

90 

14S 

27 

319 

44.40  °-5S 
°-S5 
43.91 

9.45 

3.03 

0.38 

0.22 

98 

148 

27 

320 

9.44 

3.. 58 

0.38 

0.21 

100 

147 

28 

321 

43.30 -°-5S 

9.43 

3.53 

0.38 

0.21 

103 

140 

28 

322 

42.81  °-5S 

9.42 

3.48 

0.38 

0.21 

105 

140 

29 

323 

42.27  °-5-» 

9.41 

3.43 

0.38 

0.21 

108 

145 

29 

324 

41.72  °-55 
0.54 
41.18 

9.40 

3.38 

0.38 

0.20 

III 

144 

30 

32.-) 

9.40 

3.33 

0.38 

0.20 

114 

144 

30 

32(! 

40.04-=-S-» 

9.39 

3.28 

0.38 

0.20 

117 

143 

31 

327 

40.10  °-S4 

9.38 

3.23 

0.38 

0.20 

120 

142 

32 

32S 

39.50  °-S4 

9.37 

3.18 

0.38 

0.19 

123 

141 

32 

329 

39.03  °S3 
0.54 

9.30 

3.14 

0.37 

0.19 

127 

140 

33 

330 

38.49 

9.35 

3.07 

0.37 

0.19 

130 

140 

34 

331 

37.90-°- 5  3 

9.34 

3,02 

0.37 

0.19 

1.34 

139 

34 

332 

37.43  °-S3 

9.33 

2.08 

0.37 

0.18 

138 

138 

35 

333 

30.90  °S3 

9.32 

2.93 

0.37 

0.18 

141 

137 

30 

334 

30.37  °'53 
0.52 

9.31 

2.88 

0.37 

0.18 

145 

130 

30 

335 

9.30 

2.83 

0.37 

0.18 

150 

130 

37 

33B 

35.33-°- 5^ 

9.29 

2.78 

0.37 

0.17 

154 

135 

38 

337 

34.81  °-S^ 

9.29 

2.73 

0.37 

0.17 

158 

134 

38 

338 

34.29  °-52 

9.28 

2.09 

0.37 

0.17 

102 

133 

39 

339 

33.78  -5; 
33.20 

9.27 

2.04 

0.37 

0.17 

107 

132 

40 

340 

9.20 

2.59 

0.37 

0.17 

172 

132 

41 

341 

32.75 -°-5; 

9.25 

2.54 

0.37 

0.10 

170 

131 

42 

342 

32.24  °-  ' 

9.24 

2.50 

0.37 

o.in 

181 

130 

42 

313 

31.74  °-° 

9.23 

2.45 

0.37 

0.10 

186 

129 

43 

344 

31.23  °-5' 

0.50 

30.73  „  ,„ 

9.22 

2.41 

0.37 

O.IG 

191 

128 

44 

345 

9.21 

2.30 

0.37 

0.15 

197 

127 

45 

340 

30.24-°- ^2 

9.20 

2.31 

0.37 

0.15 

202 

126 

40 

347 

2»-T*  li° 

9.20 

2.27 

0.37 

0.15 

207 

120 

40 

348 

2»-25  III 

9.19 

2.23 

0  ;7 

0.15 

213 

125 

47 

349 

2«-T6  °:S 

27.79 -"--^S  ■ 

9.18 

2.18 

0',7 

0.15 

218 

124 

48 

350 

9.17 

2.14 

0.37 

0.14 

224 

123 

49 

351 

9.10 

2.09 

0.37 

0.14 

230 

122 

50 

352 

27.31  °-''^ 
20.83  °-+^ 

9.15 

2.05 

0.37 

0.14 

230 

121 

51 

353 

9.14 

2.01 

0.37 

0.14 

242 

120 

51 

354 

20.30  °--*7 
0.47 

9.13 

1.97 

0.37 

0.14 

248 

119 

52 

355 

25.89 

9.12 

1.93 

0.37 

0.14 

255 

118 

53 

350 

25.42 -°-'*7 

24.90  °-t^ 

24.50  °-'t^ 

24.04  °-t^ 
0.46 

23.58 

9.12 

1.89 

0.37 

0.13 

201 

117 

64 

357 

9.11 

1.85 

0.37 

0.13 

208 

116 

55 

358 

9.10 

1.81 

0.37 

0.13 

274 

115 

50 

359 

9.09 

1.77 

0.37 

0.13 

281 

114 

57 

300 

9.08 

1.73 

0.38 

0.13 

288 

113 

58 

w- 


1 

•A 

Clii 

I  HI 

ma 


li 


:i       f: 


230 


T  11  K    O  U  n  I  T   ()  V   II  U  A  N  IT  S. 


TAIU-K  VI ir,  A 

lUl,   \.  —  ( 

'iinlinitiil. 

Art?. 

(tM'.O)        Diir. 

(V.H.I) 

(''■<•.  1) 

ft 

(•'.«.  2) 

(r.,-.2) 

It 

(p.r.O) 

(,...1) 

(,...•.1) 

II        It 

n 

tt 

ano 

^'''-'"^     „   ,c 

9.08 

1.73 

0.38 

0.13 

288 

113 

68 

8(11 

23.13-°^S 

0.07 

1.70 

0.38 

0.13 

29 1 

112 

58 

8ii-i 

22(19     °^^ 

9.00 

1.0» 

0.38 

0.13 

301 

111 

59 

8)13 

22.24     °-5 

9.05 

1.02 

0.38 

0.13 

308 

110 

60 

8(U 

21.80    °;^} 

9.04 

1.59 

0.38 

0.12 

310 

109 

61 

snr) 

9.03 

1.55 

0.37 

0.12 

323 

108 

62 

son 

20. 94-°-^^ 

9.02 

1.52 

0.37 

0.13 

330 

107 

63 

3(17 

20.51     °| 

9.01 

1.48 

0.37 

0.13 

338 

106 

64 

ac.s 

20.0H     °^- 

9.00 

1.45 

0.37 

0.12 

345 

105 

(15 

SCO 

19.(10  °;]j 

19.21 

8.99 

1.42 

0.37 

0.11 

353 

104 

60 

370 

8.9S 

1.39 

0.37 

0.11 

8(11 

103 

66 

.ni 

18.83 ''°-*' 

8.97 

1.30 

0.37 

0.1 1 

3(19 

102 

67 

373 

18.42     °-»' 

8.9i; 

1.32 

0.37 

0.11 

877 

101 

68 

37;J 

18.03     °-^° 

8.95 

1.29 

0.37 

0.11 

385 

100 

(19 

874 
375 

n.oa  °-*'' 

0.40 
17.23 

8.94 
8.93 

;.27 
"i.24 

0.37 
0.37 

0.11 
O.II 

393 
401 

99 
98 

70 
71 

37(> 

l(1.83-°-3'> 

8.92 

1.21 

0.37 

0.11 

410 

97 

72 

377 

10.44     °-." 

Ifi.Ofi     °-^'J 

15 -i     °-^'^ 
0.38 

15.30 

8.91 

1.18 

0.37 

0.11 

418 

9(1 

73 

871 

8..S9 

1.10 

0.37 

0.11 

420 

95 

71 

371) 
3S0 

8.88 
8.87 

1.13 
1.11 

0,37 

o.;i7 

0.11 
0.10 

435 
414 

91 
9:1 

75 
76 

8S1 

14.93 -°" 
14..',7     °-'^ 

13.85     °»^ 
13.50 

8.80 

1.08 

0.37 

0.10 

4,^2 

92 

77 

8S2 

8.85 

1.00 

0.37 

0.10 

401 

91 

7S 

3S3 

8.83 

1.04 

0.37 

0.10 

470 

90 

78 

3S4 

3M5 

f:.82 

8. SI 

1.02 
1.00 

0.37 
0.37 

0.10 
0.10 

479 

488 

89 
88 

79 
80 

3S(I 

13.15-°'" 

8.79 

0.98 

0.37 

0.10 

497 

87 

81 

3s; 

12.80     °--^3 

8.78 

0.90 

0.37 

0.10 

50(1 

8(1 

8:' 

3HS 

12.47     °-]l 

8.77 

0.94 

0.37 

0.10 

514 

85 

8' 

88  y 

12.13     III 
11. Si 

8.75 

0.92 

0.37 

0.10 

525 

84 

3!)() 

8.74 

0.90 

0.37 

0.10 

534 

82 

391 

11.48-^-" 

8.73 

0.88 

0.37 

0.10 

544 

81 

8(. 

392 

11.10    °- 

8.71 

0.87 

0..37 

0.10 

554 

80 

86 

393 

10.85     °- 

8.70 

0.85 

0.3(1 

0.10 

5(14 

79 

87 

394 

10.54   °;]; 

^0  -^     0  70 

8.(18 

0.84 

0.3(i 

0.10 

5f3 

78 

88 

395 

8.07 

0.83 

0.3(i 

0.10 

5s;i 

77 

89 

39(i 

9.93-°-^° 

8.05 

0.82 

0.3(i 

0  10 

593 

76 

90 

397 

9.(14     III 

8.(!.<; 

0  SO 

0  3(1 

0.10 

(103 

75 

91 

39  S 

9.. 35     °-  ^ 

8.03 

0.79 

0.3(5 

0.10 

013 

74 

92 

399 

'^.00 

0.7.8 

0.30 

0.10 

C23 

73 

93 

400 

«-T!^     0,, 

8.5S 

0.7T 

0.30 

0.10 

033 

72 

9.3 

401 

8..52-°- 

8.. 57 

0.70 

0.3(1 

0.10 

043 

71 

94 

403 

«-2'^     olJ 

8.55 

0.7('. 

0.30 

0.10 

653 

70 

95 

4113 

v.fls  °-  ! 

8.53 

0.75 

0.30 

0.10 

604 

69 

96 

4U4 

^■7='     L'^ 

8.51 

0.74 

0.35 

O.IO 

674 

C8 

97 

405 

7.47 

8.49 

0.74 

0.35 

0.10 

684 

67 

98 

40(! 

7  .)2-°='S 

8.47 

0.74 

0.35 

0.10 

695 

66 

98 

407 

(iMs  °-^-* 

8.40 

0.73 

0.35 

0.10 

700 

65 

99 

40  S 

fl.75     °13 

8.44 

0.73 

0.35 

0.10 

71(5 

64 

100 

409 

G.51     °-^-* 
0. 22 

8.42 

0.73 

0.34 

0.10 

727 

63 

101 

410 

0.29 

8.:!9 

0.73 

0.34 

0.10 

737 

62 

101 

411 

G.()7-°" 

8.37 

0.73 

0.34 

0.10 

.    748 

61 

102 

412 

5.85     °" 

8.35 

0.73 

0.34 

0.10 

759 

60 

10.^ 

413 

5.04     °" 

8.33 

0.73 

0.34 

0.10 

770 

59 

104 

414 

6.14     °-'^ 

0.  20 

8.31 

0  73 

0.34 

0.10 

781 

58 

104 

415 

5.24 

8.28 

0.73 

0.34 

0.10 

792 

57 

105 

410 

5.04-°-^° 
4.80     °''^ 

8.2(1 

0.74 

0.33 

0.10 

803 

56 

106 

417 

8.24 

0.74 

0.,33 

0.10 

814 

55 

107 

41S 

4.C.7     °"' 

8.21 

0.74 

0.33 

0.10 

825 

54 

107 

419 

4.33 

8.19 

0.75 

0.33 

0.10 

836 

53 

108 

420 

8.17 

0.76 

0..33 

0.10 

847 

52 

109 

TIIK   OniJIT   OF   URANUS. 


231 


T.VHLK 

VIII,  Ami.  1  —f 

iiiUniird. 

Arg. 

(h.<!.o)     Diir. 

(.....1) 

(v.v.l) 

(u.».2) 

(>K>:2) 

(,.r.0) 

(P.-.I) 

(p.C.l) 

n           tf 

ft 

n 

tf 

420 

<-''3  „,, 

8.17 

0.70 

0.33 

0.10 

847 

53 

109 

421 

4.1(1-°  'J 
4.00  °'^ 

8. It 

0.70 

0.33 

0.10 

858 

51 

110 

4'2!) 

8.11 

0.77 

o.;;3 

0.11 

870 

50 

110 

4-i:t 

8.85  °-\l 

8.00 

0.78 

0.33 

0.11 

881 

50 

111 

424 

8.0(1 

0.79 

0.33 

0  11 

893 

49 

111 

4'2.'> 

8.04 

0.80 

o.;;3 

0.11 

1)04 

48 

112 

421! 

^■*'^  oA 

8.01 

0.81 

0.31 

0.11 

010 

47 

113 

427 

3.29  °^ 

7.98 

0.83 

0.31 

0.11 

037 

40 

113 

428 

3.1(1  °\l 

7.95 

0.83 

0.31 

0.11 

938 

45 

114 

429 

3»^  o.W 

7.!I2 

0.85 

0.31 

0.11 

950 

44 

114 

430 

2.n3 

7.89 

0.80 

0.31 

0.11 

901 

44 

116 

431 

2.S2-°'" 

7.80 

0.87 

o.;io 

0.11 

073 

43 

110 

4;t2 

2.72  °''° 

7.83 

0.89 

0.30 

0.11 

984 

43 

110 

4:t;{ 

2.(12  °'° 

7.80 

0.!(0 

0.30 

0.11 

990 

41 

117 

434 

2.53  °°2 
o.o8 

2.45 

7.77 

0.93 

0.29 

0.11 

1008 

40 

118 

435 

7.74 

0.94 

0.29 

0.11 

1019 

40 

118 

4;t(J 

2.37 -°°'] 

7.71 

0.  !(5 

0.29 

0.12 

1031 

39 

119 

4;{7 

2  i)()   °o8 

7.07 

0.97 

0.29 

0.12 

1043 

38 

119 

4:i.s 

2:2.3  °°'^ 

7.04 

0. 119 

0.28 

0.12 

1054 

38 

120 

43!) 

2.1(1  °°7 
0.0s 

7.(10 

1.01 

0.28 

0.12 

1000 

37 

120 

440 

2.11     : 

7.57 

1.03 

0.28 

0.12 

1078 

30 

121 

441 

2.00 -°°S 

7.54 

1.05 

0.28 

0.13 

1089 

35 

121 

442 

2.03  °°^ 

7.50 

1.07 

0.27 

0.12 

1100 

35 

123 

443 

1.98  °°-* 

7.47 

1.09 

0.27 

0.12 

1113 

34 

123 

444 

l.!)5  °°^ 
0.03 

1.92 

7.43 

1.11 

0.27 

0.12 

1124 

33 

123 

445 

7.39 

1.14 

0.20 

0.12 

1137 

33 

123 

4411 

,  ,,,,-0.02 

7.3(1 

1.10 

0.20 

0.12 

1149 

32 

123 

447 

1.88  °°' 

7.32 

1.18 

0.20 

0.12 

1100 

32 

124 

44S 

1.87   °°' 

7.28 

1. 21 

0.20 

0.12 

1172 

31 

124 

44!) 

1.87  °'°° 
0.00 

7.24 

1.23 

0.25 

0.12 

1184 

30 

125 

4:.o 

1.87  , 

7.20 

1.20 

0.25 

0.12 

1190 

no 

125 

451 

1.88+°'°' 

7.10 

1.28 

0.25 

0.12 

1208 

30 

125 

453 

l.!l()  °°- 

7.12 

1.31 

0.24 

0.12 

1220 

29 

120 

453 

1.92  "■°-' 

7.08 

1.33 

0.24 

0.12 

1 232 

28 

120 

454 

0.03 
1.98  , 

7.01 

1.30 

0.24 

0.13 

1243 

28 

127 

455 

7.00 

1.39 

0.23 

0.12 

1255 

28 

127 

45(; 

2.01 +°°.^ 

0.90 

1.42 

0.23 

0.12 

1207 

27 

127 

457 

2  0(1  °°-^ 

0.91 

1.45 

0.23 

0.12 

1279 

27 

128 

45H 

2.11  °°3 

0.87 

1.47 

0.22 

0.12 

1291 

20 

128 

45!) 

2.1(1  °°^ 
0.07 

2.23 

0.83 

1.50 

0.23 

0.12 

1302 

20 

128 

4(->0 

0.78 

1.53 

0.22 

0.12 

1314 

20 

128 

4(i! 

0.74 

1..50 

0.21 

0.12 

1320 

25 

129 

402 

0.(19 

1.59 

0.21 

0.12 

1338 

25 

129 

4(13 

2.44  °'°7 

0.0.  > 

1.02 

0.21 

0.12 

1350 

25 

129 

4(14 

2.53  °°'^ 
0.09 

2-''2  ,„„„ 

0.00 

1.05 

0.20 

0.12 

1301 

25 

130 

4f,5 

0.50 

l.fiS 

0.20 

0.12 

1373 

24 

130 

4r>a 

2^1+0.00 

0.51 

1.72 

0.20 

0.12 

1385 

24 

130 

4fi7 

2«2  °-  ! 

0.40 

1.75 

0.19 

0.13 

1390 

24 

130 

4fiS 

2'92  0 : 

0.41 

1.78 

0.19 

0.13 

1408 

24 

130 

4(19 

304  ::;-; 

0.37 

1.81 

0.19 

0.13 

1420 

23 

131 

■170 

3.15 

0.32 

1.84 

0.18 

0.13 

1431 

23 

131 

471 

•J  .)s+°'.'5 

0.27 

1.88 

0.18 

0.13 

1  1 13 

23 

131 

472 

3.41  °'-^ 

0.22 

1.91 

0.18 

0.12 

1 455 

23 

131 

473 

3.. 55  °'4 

0.17 

1.94 

0.17 

0.12 

1400 

23 

131 

474 

3.(19  °'-* 
0.14 

3.83  , 

0.12 

1.98 

0.17 

0.12 

1478 

23 

132 

475 

0.07 

2.01 

0.17 

0,12 

1489 

23 

132 

47G 

3. 98+°- '5 
4.14  °'^ 

0.02 

2.04 

0.1(1 

0.12 

1501 

23 

132 

477 

5.90 

2.07 

0.10 

0.12 

1512 

23 

132 

478 

4.31  °;7 

5.91 

2.11 

0.10 

0.12 

1524 

23 

132 

479 

4.48  °'7 
0.17 

4.05 

5.80 

2.14 

0.15 

0.12 

1535 

23 

132 

480 

5.81 

2.18 

0.15 

0.12 

1540 

23 

132 

iMii: 


.i  * 


)■    H 


i  IP 


f 


i     i! 


ii  ;! 


233 


THE   ORBIT  OF   URANUS. 


TABLK 

VIII,  A 

KO.  1. —  ContinvriK 

A.V'. 

(f.c.O)   Diff. 

(v.x.l) 

(•'•'•.  1) 

(U.S.2) 

(o.r.2) 

Cp.c.o; 

(f)."!) 

(P.e.l) 

If           II 

tt 

$t 

ft 

ft 

480 

-;Uo..8 

5.81 

2.18 

0.15 

0.12 

1540 

23 

133 

4Sl 

5.75 

2.-1 

0. 15 

0  12 

1558 

23 

132 

4,S2 

5«=2   nl^ 

5.70 

2.24 

0.15 

0.12 

1509 

23 

133 

483 

5.21  ""^ 

5.05 

2.28 

0.14 

0.12 

1 5S0 

23 

132 

484 

^•^"  .r^ 

5.59 

2.31 

0.14 

0.12 

1591 

23 

133 

485 

5. CI  , 

5.54 

2.35 

0.14 

0.12 

!  002 

24 

133 

4Si; 

5.81+°" 

5.48 

2  38 

0.13 

0.12 

1013 

24 

133 

487 

fi.03  °- 

6.43 

2.42 

0.13 

0.12 

1 024 

24 

133 

48S 

f'-'-^l  o. 

5.37 

2.45 

-.1.13 

0.11 

10,35 

24 

134 

481) 

6.70  , 

5.32 

2.48 

0.12 

O.ll 

1040 

25 

134 

490 

5.20 

2.52 

0.12 

0.11 

1057 

25 

133 

491 

f;.!l3+°-'-5 

5.20 

*2.55 

0.12 

0.11 

1008 

25 

133 

492 

7.17  °-^-t 

5.1;-) 

2.59 

0.12 

0.11 

1079 

25 

133 

493 

7.41  "•'■> 

5.119 

2.02 

0.11 

0.11 

1090 

25 

133 

494 

7.(!(!  °%5 

O.  20 

6.U3 

2.00 

0.11 

0.11 

1700 

20 

133 

495 

7.92  ,  '  . 
8.1,S+°-^^ 
8.-I4  °-^^ 

4.98 

2.09 

O.ll 

0.11 

1711 

27 

133 

49G 

4.92 

2.72 

0.11 

O.ll 

1721 

27 

133 

49T 

4.80 

2.70 

0.10 

0.11 

1732 

28 

133 

498 

8.71  °H 

4.80 

2.79 

O.IO 

0.11 

1742 

28 

l.i3 

499 

8.99  ""-'^ 
0.27 

9  20  ^  , 

4.75 

2.83 

0.10 

0.11 

1753 

28 

133 

500 

4.09 

2.80 

0.10 

on 

1703 

29 

133 

501 

9.55+°-^'^ 

4.03 

2.89 

0.10 

o.ll 

17-4 

30 

133 

502 

9.84  °-'' 

4.57 

2.93 

0.09 

0.10 

'784 

30 

133 

503 

10.13  °'^'^ 

4.51 

2. 91'. 

0.09 

0.1 1» 

1794 

31 

133 

604 

10.43  °'^ 
10.74  , 

4.44 

2.i'9 

0.09 

0.10 

1804 

31 

133 

505 

4.39 

3.02 

0.09 

0.10 

1S14 

32 

133 

&0I> 

ll.0;-.+°'^' 

4.33 

3.0;! 

0.09 

0.10 

1824 

32 

133 

507 

11. 3(,  °'' 

4.28 

3  09 

0.08 

0.10 

1834 

33 

133 

508 

11.  (!8  °y- 

4  22 

3.12 

0.08 

(1. 10 

1844 

34 

133 

509 

12.00  '"■•'^ 
°J3 

4.10 

3.15 

0.08 

0,10 

1S53 

35 

133 

510 

12.33  , 

12.r,.!+°-|3 

4.10 

3.18 

0.08 

0.09 

1 803 

!>;"» 

133 

511 

4.04 

3.21 

0.08 

0.09 

1873 

30 

133 

5i: 

13.00  °34 

8.93 

3.25 

0.07 

0.09 

lss2 

37 

133 

;-.l:i 

13.34  l\\ 

3.92 

3.28 

0.07 

0.  19 

1892 

38 

133 

614 

13.G8  °;j^^ 

'4.03  , 
14.39+°'^" 
14.75  °'^^' 
15.11  °-|^ 

8.80 

3.31 

0.07 

0.09 

1901 

39 

133 

515 

3.S0 

3.34 

0.07 

0.09 

1910 

40 

133 

510 

3.74 

3.37 

0.07 

0.09 

1920 

40 

133 

517 

3.08 

3.10 

0.07 

0.09 

1 929 

41 

133 

518 

3.02 

3.13 

0.97 

0.09 

1938 

42 

133 

519 

3.50 

3..<i6 

0.07 

0.08 

l'.M7 

43 

133 

520 

3. 50 

3.49 

0.00 

0  08 

1 9,'^„i 

44 

132 

521 

3.41 

3.51 

0.00 

O.CM 

1904 

45 

132 

522 

3.38 

3.54 

0.00 

0.08 

1973 

40 

132 

523 

1C..99  °'5 

3.32 

3.57 

O.Oi! 

0.08 

1982 

47 

132 

524 

17.77  , 

3.27 

3.00 

0.06 

0.08 

l'J;»0 

48 

132 

525 

3.21 

3.02 

0.06 

0.08 

1999 

49 

132 

52fi 

18.17+°'° 

3.15 

3.0."i 

0.06 

0.08 

2007 

50 

l:;2 

527 

18.07  °-t° 

3.09 

3.07 

0.00 

0.08 

2010 

51 

132 

528 

18.97  °-'° 

3.03 

3.70 

0.06 

0.07 

2024 

52 

1 32 

529 

19.38  °-»' 
0.41 

19.79  , 

2.97 

3.73 

0.05 

0.07 

2032 

53 

132 

530 

2.91 

3.7.'> 

0.05 

,  0.07 

2040 

55 

132 

531 

20.21 +°-'' 

2.  MO 

3.77 

1«.05 

"0.07 

2048 

50 

132 

532 

20.(12  °-" 

2  80 

3.  so 

0.05 

0.07 

2050 

67 

132 

533 

21.05  °« 

2.75 

3.82 

0.(i5 

0.07 

2(103 

58 

132 

534 

21.47  °-»' 
0.43 
21.90  , 

2.09 

3.84 

0.05 

0.07 

2071 

69 

131 

535 

2.03 

3.87 

0.05 

O.OT 

2079 

01 

131 

531) 

.,J  ;,4+o-14 

2.57 

3.8i) 

0.05 

0.07 

2080 

62 

131 

537 

"VT  °-!^ 

2.52 

3.91 

0.05 

0.07 

2094 

03 

131 

538 

23.21  °^* 

2.40 

3.93 

0.05 

0.07 

2101 

04 

131 

539 

23.. IC,  "■'•S 
0.44 
24.10 

3.40 

3.90 

0.05 

0.00 

2108 

66 

;  131 

1 

540 

2.35 

3  OS 

0.05 

0.00 

2115 

67 

131 

THE   ORBIT   OF   URANUS. 


233 


TAHLK 

VIII,  A 

10.  1. —  Concluded. 

Aig. 

(o.e.O)       Diff. 

(u.s.l) 

(...c.l) 

(0.8.2)    ] 

0>.6'.2) 

(p.c.O) 

{p.»l) 

(p.o.l) 

ft                       ff 

fr 

ft 

// 

640 

2*10.      ,A 

2.35 

3.98 

0.05 

0.00 

2115 

07 

131 

f)41 

2.2'J 

4.00 

C.')5 

0.00 

2122 

08 

131 

542 

2.24 

4.02 

0.05 

0.00 

2129 

70 

131 

643 

2.18 

4.04 

0.05 

0.00 

2135 

71 

130 

644 

2.13 

4.C.' 

0.05 

O.OG 

2142 

73 

130 

545 

2.08 

4.07 

0,05 

0.06 

2149 

74 

130 

51() 

2.02 

4.0!) 

0.05 

0.00 

2155 

75 

130 

547 

27.;i2    °-^^ 

l.i)7 

4.11 

0.05 

0.00 

2102 

77 

130 

54S 

2«-2T  o:£ 

1,'J2 

4.13 

0.05 

0.00 

2108 

78 

130 

l>i'J 

1.87 

4  14 

0.05 

0.06 

2174 

80 

130 

550 

2S.75          o 

1.H2 

4.',; 

0.05 

0.00 

2180 

81 

130 

551 

2!t.2.i+°-*° 

1.77 

4  1: 

0.05 

0.00 

2180 

83 

130 

652 

21)71     0-48 

1.72 

4,19 

0.00 

0.00 

2191 

84 

130 

653 

30.20     °-»9 

1.07 

4.20 

0.00 

0.00 

2197 

8". 

130 

564 

30.0'J     °-'9 
0.49 

1.C2 

4.i« 

0.00 

0.00 

2203 

1 

130 

555 

31.18  , 

1.57 

4.23 

0.00 

0.00 

2208 

89 

1.30 

550 

31.07 +°-»9 

1.52 

4.25 

0.00 

0.00 

2214 

90 

130 

557 

32.10     °-*'} 

1.48 

4.20 

0.00 

0.00 

2219 

92 

1:^0 

558 

32.00     °-5° 

1.43 

4.27 

0.00 

0.00 

2224 

94 

130 

551) 

3U.j;    "50 

.-!.•!. 07  .      , 

1.38 

4.28 

O.oG 

0.00 

2229 

95 

130 

SCO 

1.34 

4,29 

0.00 

0.00 

2234 

97 

i;io 

5(il 

34. 17+°- 5° 

1.2!) 

4,31 

coo 

0.00 

2238 

98 

l:w 

5(>-i 

34.08     "  5' 

1.25 

4.32 

0.07 

0.00 

2213 

100 

130 

5(i.'i 

35.1'..     "^^ 

1.20 

4.32 

0.07 

0.00 

2248 

101 

130 

5(;4 

35,70     °-^; 

1.10 

4.;i3 

0.07 

0.00 

22.'!  .i 

103 

130 

5i;5 

1.12 

4.34 

0.07 

0.0() 

2250 

105 

130 

5or> 

30.73+°-^ 

1.08 

4.35 

0.07 

(..'<; 

2201 

100 

130 

507 

37.25     °-5^ 

1.03 

4.:!0 

0.07 

0.00 

2205 

108 

130 

50S 

37  70     °-^ 

0.!)'.) 

4  37 

0.07 

0.00 

2209 

110 

130 

50'.) 

3s.2',t  °-:- 

o.;if; 

4.38 

0.07 

0.00 

2272 

111 

130 

570 

38. SI    ,       ^ 

0.02 

4,38 

0.07 

0.00 

2270 

113 

130 

571 

3!..3»-*-°-''-5 

0.88 

4.39 

0.(17 

0.00 

2280 

115 

130 

572 

3U.S0     °-^^ 

0.84 

4.40 

0.07 

0.07 

2283 

110 

130 

57:5 

40.3!)     °-^^ 

0.80 

4.10 

0.08 

0.(17 

22x7 

118 

130 

574 

40.:.2    :^ 

0.77 

4,41 

0.08 

0.07 

2290 

120 

131 

575 

0,73 

4,41 

0.08 

0.(17 

2293 

121 

131 

5 10 

41.!)8+°-^:-^ 

0,70 

4,(2 

0.08 

0.07 

22:10 

123 

131 

577 

42.,^2     °-   1 

0.i;7 

4,(2 

0.08 

0.07 

22;i9 

125 

131 

57  S 

43.05     °^ 

0.03 

4.42 

0.08 

0.07 

2302 

120 

131 

579 

«-^'»  :ii 

0.00 

4.43 

0.08 

0.07 

2304 

128 

131 

5S0 

44.12 , 

0.57 

4,43 

0.08 

0.07 

2.307 

130 

131 

5S1 

44.00 +°-"!t 

0,5  4 

4.13 

0.08 

0.08 

23(19 

132 

131 

5.S2 

45.20     "-y^ 

0,51 

4,(4 

0,(19 

0.08 

2312 

l:!3 

131 

5s:i 

45,71     °5» 

0,4  8 

4,(1 

0,(19 

o.os 

23 1  I 

135 

131 

5S4 

40.28     °?» 

0.40 

4,44 

0.09 

0.08 

2310 

i;{7 

132 

5S5 

40.83 

0.43 

4,44 

0,09 

0.08 

2318 

138 

132 

5S0 

47.37 +°-^> 

0,40 

4.4) 

0.09 

0.08 

2319 

140 

132 

5.S7 

47,1)1     °5^ 

0.38 

4.!. 

0.09 

0.09 

2321 

142 

l:!2 

5SS 

48.10     °-5.> 

0.30 

4,4i 

0.09 

0.09 

2322 

144 

1:12 

589 

49.00     °5^ 
40.55  , 

0.33 

f  '■■( 

0,09 

0.09 

2324 

145 

132 

500 

0.31 

■' ,  i  i 

0,09 

0,09 

2325 

147 

132 

5 '.11 

50, 10+°  5' 

0.20 

4.41 

009 

0.09 

2320 

I '9 

132 

5!)2 

50,0  4     °-5> 

0.27 

4.44 

0,10 

0.09 

2327 

150 

133 

5li;J 

51,10     °5.S 

0.25 

4.44 

0,10 

0.10 

2328 

152 

133 

5i»4 

51,74     °ll 
52.20  , 

0,j3 

4.43 

0,10 

0.10 

2329 

154 

133 

5!>5 

C  .!2 

4.43 

0,10 

0.10 

2330 

155 

133 

5!)0 

52.s4+°- 

O.jl 

4.43 

0.10 

O.li) 

2330 

1.^7 

133 

5117 

53,38     °.';4 

O.H 

4.43 

0,)0 

0.11 

2330 

159 

l:i3 

5'.18 

53, '.)3     °-^5 

0,17 

4.43 

0,10 

0.11 

23:!  1 

101 

134 

5il!» 

51.4S     °?5 
0-5S 
55.03 

0  lO 

4.42 

O.IO 

0.11 

2:131 

102 

l:!t 

000 

0.14 

4.42 

0.10 

on 

23:tl 

104 

i;u 

it 


30      July.  1873 


23  i 


THE   ORBIT  OF   URANUS- 


t  ;li 


(., 


I!!i 


TAHLR  IX,  Ami.  2— 

VCI'ION 

oi-  Saturn. 

1 

Arg. 

(c.r.O)     Dill. 

1 

(i.'.s.l)  Dili.  Sco.var. 

{r.c.D  Dili,  g 

L'c.var. 
It 

(r.*i.2)  Diff.  S 

t'c.var. 

(v.u.2)  Dill".  Sec.var. 

II           II 

//                      // 

/' 

//          // 

/' 

//           /' 

" 

0 

38.54  , 

139.94  , 

2.53 

293.78  , 

0.50 

182.02  ,        „ 

1.77 

244.07 

0.12 

1 

S8.63+°-°9 

141.38  +  ;-'| 

2.52 

293.81 +°°3 

0.55 

183.80  +  '-'" 

1.70 

244.10-°-57 
243.52     °-f^ 

0.12 

2 

38.72     °-°5 

142.82       -^l 

2.50 

293.83+°-°^ 

0.54 

184.90      '-'^ 

)  74 

0.12 

a 

38.81     °°9 

144.26     '-^^ 

2.49 

293.83     °°° 

0.53 

180.12     '■'" 

1.73 

^^     I 

o.u 

4 

38.90     °°l 

145.70     '•'»■* 

2.48 

293.82-°°' 

0.52 

187.27     '-'5 

1.71 

0.11 

0.08 

1.44 

0.03 

'■'5 

5 

38.98  , 

Kf.14  ... 

2.47 

293.79     „„. 

0.51 

188.42  , 

1.70 

0.11 

(i 

39.07+°°'^ 

148.58+   -y 

2.40 

293.75-°-°:^ 

0.50 

189.50 +''-^ 

1.09 

O.ll 

T 

39. 1(!     °-°l 

150.01     1  •♦•? 

2.45 

293.70     °°^ 

0.49 

190.09      '-'^ 

1.67 

0.11 

8 

39.24     °°l 

1-'1«        11 

2.44 

29,3.03     °-° 

0.48 

191.82     ''3 

l.Ofi 

0.10 

» 

39.33     '■'■°'> 

152.89       -^j 

2.43 

293.54     °°^ 

0.47 

192.94     '■'- 

1.04 

0.10 

o.oS 

>-4J 

0.10 

I. II 

10 

39.58     °°^l 

154.32 

2.43 

293.44 

0.40 

194.05 

1.03 

23S.48         .„ 
237.sO-°-f 
2.-J7.11     °-^y 

0.10 

11 

155.7(!  +  '->-> 

2.41 

293.33-°" 

0.15 

195.10  +  '" 

1.01 

0.10 

1:2 

157.19     '•^■5 

2.40 

293.20     °'-' 

0.44 

190.20     ■•'° 

1.00 

0.10 

13 

39.C7     °-°J, 
39.75     °°^ 

158.(i2     ■•« 

2.39 

21)3.00     °'j 
292.90     °''' 

0.43 

197.35     '"^ 

1.58 

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0.10 

14 

100.05     '•■♦•> 

2.38 

0.42 

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1.57 

235.09     °-7- 

0.10 

o.og 

'•43 

0.17 

1.08 

0-73 

15 

39.84  ,„o 
39.92+°°^ 

1(51.48  , 

2.30 

292.73          . 

0.42 

199.51  , 

1..55 

234f»P     „  ,, 

0.10 

11; 

102.90  + '••*^ 

2.35 

0.41 

200.58  +  '-°J 
201.04     '"^ 

1.53 

234.22-°-   t 

0.09 

IT 

4»-0»     Irl 

104.32     '■•♦^ 

2.34 

292!3"t     °--' 

0.40 

1.52 

2;!3.48     °-7-* 

0.09 

IS 

^o-ic    III 

105.74     '•'^ 

2.33 

292.13     °-' 

0.39 

202.09      '■°5 

1.50 

232.73     °-75 

0.09 

19 

107. IG     '••*^ 

2.32 

291.90     °-'i 

0.38 

203.74     '"S 

1.49 

231.90     °-'^ 

0.09 

0.09 

1.42 

0-^5 

1.04 

0.78 

20 

--+:■- 

108.58 

2.31 

291.05           , 

0.37 

204.78  , 

1.47 

231.18 

0.09 

21 

109.99 +  '•■♦' 

2.30 

21)1.39-°-^'^ 

0.30 

205.sl+'-°3 

1.40 

23().;i9-°-?'^ 

0.09 

22 

-"^  H 

171.40     '■->' 

2.29 

2:)1.12     °-'7 

o..;5 

200.  S3     '•°- 

1.44 

229.  CO     °-l'^ 

0.09 

2;{ 

172.81     '  •*• 

2.27 

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0.35 

207. S4     '■°' 

1 .  13 

22s.  SO     °-f 

0.l(» 

24 

*^-^^       °nn 

174.22     '•^' 

2.2>> 

290:53     °-.'° 

0.34 

208.85     '■°' 

I   11 

227.98     °-'^' 

0.10 

0.09 

1.40 

0.31 

1.00 

0.83 

2J 

-:-+o:^ 

175.02  ,.   . 

2  25 

290.22 

0..33 

209.85  , 

1.40 

227.15     ^„ 

0.10 

2(5 

177.02+'-'° 

2.24 

289.89- °"-'' 

0.32 

2I0.84+°-W 

1.39 

220.31 -°-''» 

O.IO 

27 

4tJ-«9     0°^ 

178.42     '-t" 

2.23 

289.,55     °-5;f 

0.31 

211.S2     °-'>^ 

1.37 

225.40     °if 

0.10 

2s 

^"•'•"^     000 

!  79.81      '••''> 

2.21 

289.19     °--''' 

0.30 

212.79     yi 

l.3(; 

224.00     °-^ 
S23.74     °it 

0.11 

29 

4107  12 

;sl.20    ■•■''> 

2.20 

2^8.82     °-" 

0.30 

213.70     yi 

1.34 

0.11 

0.09 

>  39 

0.38 

°'J5 

0.87 

:iO 

^'^S+o:^ 

1S2.59  , 

2.19 

2SS.44 

0.29 

214.71  , 

1.33 

.).)0  ,;)7 

0.11 

31 

183.117 +'--^^ 

2. IS 

2SS.04-°-»° 

0.2S 

215.00 +°-9S 

1.32 

221:09-°-^ 

0.11 

;i2 

*»:i5    °'° 

1S.V35      '••'' 

2.1; 

2S7.(;3    °-*' 

0.2s 

210.00     °'^-^ 

1.30 

221.10     °-'''^ 

0.11 

;!:} 

^'•4*      o,o 

180.73     '-3^ 

2.15 

-S7,20     °-^^ 

0.27 

217.53     °'^^ 

1.29 

220.20     °-*^° 

0.12 

34 

^l-Sl       on 

1S8.10         " 

2.14 

2S0.70     °-^j 

0.20 

21S.44     y\ 

1.27 

219.28     °-'^' 

0.12 

0.   10 

'•37 

0.4O 

o.yl 

0.92 

35 
3(> 

"°'>*r: 

H9.47  ,  ,     r 
190.  S3 +  '•■''• 

2.13 
2.11 

si;-°-^'^ 

0.20 
0.25 

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1.20 
1.25 

21s. 30     „     , 
2 17. 43 -°-'-'-' 

0.12 
0.12 

37 

"">     oZ 

192.19     '-'^ 

2.10 

2S5.35     °-''^ 

0.24 

.,.., ,,  °-^') 

1.23 

2IC.49     °'>^ 

0.12 

3S 

:i:s  B 

193.55      •••'? 

2.09 

2S4  80     °j 

0.23 

^'^■■'^     0.86 

1.22 

215.55  yj 

214.59     °-''^ 
0.97 

0.13 

39 

194.90         1: 

'33 

2.07 

2S,.35     m 

0.23 

1.20 

0.13 

40 

^B 

190.25  , 

2.00 

2.S3.S4 

0  22 

223.70  ,      „ 
•>■>,  ci  4-0.85 

.1.19 

213.02 

0.13 

41 

197.59 +  '■•'■* 

2.1(5 

283.31 -°-53 

0.22 

1.18 

212.05 -°-''7 

0.13 

42 

198.93     '  ■'"* 

2.03 

2S2.70     °-55 

0.21 

225.45     °-^^ 

1.10 

211  07     °''"'* 

0.14 

43 

*'^*i     o.\, 

200.20     '■■'■' 

2.(12 

2S2.20     °'5 

0.21 

■).),;  .>^      0-3 

1.15 

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0.14 

44 

4^.55     rt  , . 

201.59      '•■'•' 

2.00 

281.03     °-57 

0.20 

227;ll     °-^J 

1.13 

209.08      '■°° 

0.14 

0. 1 1 

'•32 

°S') 

o.bi 

1. 01 

45 

42.98     „.,, 
43.09     0.,, 

202.91  . 

1.99 

2>*1.04 

0.20 

227  02 
^•',^72+°-''° 

1.12 

20s.  07 

0.!5 

4(1 

204.23  +  ''^* 

l.'.i7 

280.41-°-^'° 

0.19 

1.10 

207.00-'°' 

0  15 

47 

205.54     '5' 

1.90 

279.S2     °';' 

0.19 

229.51     °-''> 

2:!(i.29     °-7';^ 

231.07    °■7^^ 
0.76 

l.((9 

200.04      '■°' 

0.15 

48 

200.  S5     '■■^' 

1    14 

2T9.20     °-^^ 
27S.5G     °^-> 

O.H 

l.((7 

205.01      ""-i 

0.15 

49 

208.15      '    '° 

1  93 

0.18 

1.00 

204.57      '•°-' 
••°5 

0.10 

50 

43.20  , 

209.44 

1.92 

277.91         , 

0.17 

231. H3  ,^  ,. 

1.04 

203.52 

0.10 

51 

43.32 +°'  = 

210.73 +'■*:> 

1  91 

277.24 -'^f  7 

0.17 

232.58+°- 75 

1.(13 

2((2.17-'°P 

0.17 

52 

"•«     0',: 

212.01      '•-'„ 

1.89 

270.50     °'!^ 

((.10 

233.32     °-7» 

l.((l 

201,41      '•°'^' 

0.17 

53 

43.54     °-    ' 

213.29     '  ^;^ 

1.88 

2V-..87     °-^'> 

0.10 

23l.((5     °-73 

1.(10 

200.34      '■°l 

0.18 

54 

43.05     °" 

214.57      ' 

1.87 

275.17     °-7° 

O.lfi 

234.77     °l' 

0.99 

1911.20      '"'^ 

0.18 

0.12 

1.27 

0.72 

0.71 

1.08 

55 

43.77  , 

215.^4 

1.H5 

271.45 

0. 15 

235.48  ,      , 

0.97 

19S  IS 

0.19 

5(! 

«•"«+!■  1 

217.10  +  '-^'^ 

1.84 

273.72-°-" 

0.15 

2;i0.l8+°J° 

230.H7     °p 

237.51     °f7 

23<.20     °  '" 
0.(15 

0.90 

197.09-'°'^ 

0.20 

57 

44.00  °- 

2H..I5     ''^S 

1.82 

27' 98     °'''* 

0.15 

0.95 

19C.((0      '•°'' 

0.2(( 

5S 

44  12     °" 

21 9. CO      '  '^ 

l.Hl 

o-T)'.,.!     0.75 

0.15 

0.94 

191.90      '■'° 

0.21 

59 

44.21     °" 

220. '<4     '"^ 
■  -3 

1.79 

27i:4o  °;:; 

O.lt 

((.92 

1:13.79      '" 
1. 1.> 

0.21 

CO 

44.35 

2:;2.il7 

1.7S 

27((.C9 

0.11 

.»;;■.;   S.'» 

((.91 

192  >;7 

(i..'2 

THE    ORBIT   OF   URANUS. 


235 


TARLE  IX, 

Aro.  2 

.  —  Con 

billed. 

Arg. 

(u.«.3) 

(i).c.3) 

(U.8.4) 

(U.C.4 

(p.c.O) 

(p.s.l) 

(p.C.l) 

(p.s.2)  (p.c.2 

(p.K.3) 

(p.c.Sj 

0 

11.40 

ft 

14.53 

tr 
1.58 

// 

1078 

182 

1513 

711 

306 

137 

85 

1 

11.52 

14.48 

1.59 

1.66 

1079 

182 

1526 

711 

303 

137 

85 

9 

U.fil 

14.43 

1.00 

1.05 

1080 

183 

1539 

710 

293 

137 

85 

3 

11.75 

14.38 

1.01 

1.04 

1080 

184 

1551 

709 

295 

137 

85 

4 

11.87 

14.33 

1.62 

1.03 

1681 

185 

1504 

70  J 

291 

137 

85 

5 

11.98 

It. 28 

1.03 

1.62 

1682 

187 

1.577 

707 

288 

137 

85 

6 

12.10 

It. 22 

1.04 

1.01 

1083 

188 

1590 

700 

284 

137 

85 

7 

12.21 

14.10 

1.04 

1.00 

1083 

190 

1003 

704 

281 

138 

84 

8 

12.32 

14.10 

1.05 

1.50 

1083 

191 

1015 

703 

278 

138 

84 

9 

12.43 

14.04 

1.66 

1.58 

1682 

193 

1028 

701 

274 

138 

84 

10 

12.54 

13.97 

1.07 

1.50 

1082 

195 

1641 

700 

271 

138 

84 

11 

12.05 

13.91 

1.68 

1.55 

1081 

197 

1054 

099 

267 

138 

84 

12 

12. 7G 

13.84 

1.09 

1.54 

1080 

199 

1006 

09  ( 

264 

138 

84 

l:i 

12.87 

13.77 

1.09 

1.53 

1079 

201 

1079 

090 

260 

138 

84 

U 

12.97 

13.70 

1.70 

1.52 

1078 

203 

1091 

095 

257 

137 

83 

15 

13.07 

13.03 

1.71 

1.50 

1077 

200 

1704 

691 

253 

137 

83 

|{> 

13.18 

13.50 

1.71 

1.49 

1075 

20  < 

1710 

692 

250 

137 

83 

IT 

13.28 

13.49 

1.72 

1.48 

1074 

211 

1729 

091 

240 

137 

83 

IS 

13.38 

13.41 

1.73 

1.47 

1072 

21  1 

1741 

6-19 

243 

137 

82 

19 

13.48 

13.33 

1.74 

1.45 

1070 

217 

1754 

087 

239 

137 

82 

20 

13.58 

13.25 

1.74 

1.44 

1008 

220 

1700 

085 

230 

137 

82 

:>1 

13.(18 

13.17 

1.75 

1.43 

1000 

223 

1778 

0^'3 

233 

137 

82 

•2i 

13.78 

13.09 

1.75 

1.42 

1004 

227 

17V0 

681 

230 

137 

81 

2:j 

13.S7 

13.01 

1.70 

1.40 

1002 

230 

n02 

07.1 

220 

137 

81 

24 

13.97 

12.92 

1.7G 

1.39 

1000 

23  4 

1S14 

C77 

22J 

137 

81 

2r) 

14. on 

12.83 

1.77 

1.37 

1058 

2:!7 

1 820 

075 

220 

137 

81 

2fi 

14.15 

12.74 

1.78 

1.30 

1050 

241 

ls:!s 

073 

217 

137 

81 

27 

U,2t 

12.05 

1.78 

1.35 

1053 

245 

Is5(t 

071 

214 

137 

80 

2S 

14.33 

12.50 

1.79 

1.34 

1051 

219 

ls(;2 

00  i 

211 

137 

80 

29 

14.41 

12.47 

1.79 

1.32 

1048 

253 

1874 

GOO 

208 

137 

80 

no 

11.50 

12.37 

1.79 

1.31 

1045 

258 

issi; 

004 

205 

137 

79 

;ti 

14.5S 

12.28 

1.S0 

1.30 

1042 

202 

189s 

002 

202 

137 

78 

3J 

H.c.r, 

12.19 

1.80 

1.28 

10.38 

2i;7 

1910 

051) 

190 

137 

78 

3:1 

14.74 

12.09 

1..S0 

1.27 

1035 

272 

1922 

057 

190 

137 

78 

34 

14.82 

11.99 

1.8.;. 

1.25 

1031 

277 

lOiU 

054 

193 

137 

78 

3.") 

11.90 

11.80 

l.sl 

1  24 

1027 

2S2 

1940 

052 

no 

137 

77 

,'<i; 

H.97 

11.79 

1.81 

i   "'2 

ir,23 

2-^7 

1'.l5-t 

050 

187 

137 

77 

37 

15.  ((5 

11.09 

1.81 

1  21 

1 1;  1 9 

^'.>-2 

1970 

647 

184 

137 

77 

3S 

15.12 

11.59 

1.82 

1.20 

1015 

297 

19«2 

045 

isl 

137 

70 

39 

15.19 

11.49 

1.82 

1.18 

1011 

3tt2 

1993 

042 

179 

137 

76 

40 

15. 2n 

11.38 

1.82 

1.17 

lOOS 

30S 

201 15 

Olii 

170 

137 

70 

41 

15.33 

11.28 

1.S2 

1.15 

1002 

313 

20 1<; 

037 

17  ! 

137 

70 

42 

15.39 

11.17 

1.82 

1.11 

1597 

319 

2028 

035 

171 

137 

70 

43 

15.40 

11.  oo 

1.82 

1. 13 

1592 

325 

2039 

032 

los 

137 

75 

44 

15.52 

10.95 

1.82 

1.11 

1587 

3.31 

2051 

030 

160 

137 

74 

4.'> 

15.58 

10. St 

1.S2 

1.10 

1 582 

337 

2003 

027 

103 

1.37 

74 

4(i 

15.(;4 

10.,.; 

1.82 

1.0S 

1577 

313 

2073 

0  2  4 

100 

137 

74 

47 

15.70 

10.02 

1.82 

1.07 

1572 

349 

20S4 

021 

158 

137 

73 

4S 

15.7C 

10.51 

1.82 

1.00 

1500 

355 

2095 

0, 1 0 

155 

137 

73 

49 

15. Ml 

10.40 

1.82 

1.04 

1501 

302 

2100 

0 1  0 

153 

137 

73 

r>o 

15.86 

10.28 

1.S2 

1.03 

1550 

308 

2117 

0  1  ■ 

150 

137 

72 

61 

15.91 

10.17 

l.x-i 

1.01 

1550 

375 

2128 

r,i(i 

148 

137 

!  72 

5ii 

15.90 

10.05 

1.82 

1.00 

1545 

381 

213S 

007 

145 

137 

72 

r)3 

10.01 

9.94 

l.sl 

0.08 

1539 

388 

21  19 

00  4 

H3 

137 

72 

f)4 

10.05 

9.82 

1.81 

0.97 

1533 

394 

2159 

001 

140 

137 

72 

55 

10.09 

9.70 

1.81 

0. 90 

1527 

401 

2170 

598 

13S 

137 

■  71 

5n 

10.13 

9.59 

1.81 

0.94 

1521 

408 

2180 

595 

135 

137 

71 

57 

10.17 

9.47 

1.80 

0.93 

1515 

415 

2191 

592 

133 

137 

71 

58 

10.20 

9.35 

1.80 

0.92 

150!) 

422 

2201 

589 

1.11 

137 

71 

59 

10.23 

9.23 

1.80 

0.90 

1502 

430 

2212 

5, SO 

129 

137 

71 

1 

fiO 

10.20 

9.11 

1.80 

0.89 

1490 

437 

2222 

683 

127 

137 

1  '' 

236 


THE   ORBIT   OP   URANUS. 


TAIJLE  IX,  Ami.  2.— Continued. 


\rK.  '(w.c.O)    Diir.  (r.».l)  Diir.  Sccvur.  (I'.f.l)  DilT.  Stcvur'!  u.«.2)  Diff.  Sec.vnr, 


(il 
«2 
f,3 
C4 

C5 
Gl> 
(!7 
68 
69 

70 
71 
7-2 
73 
74 

75 
7t! 
77 
78 
7  it 

80 
81 
82 
83 
84 

85 

8<; 

87 
8S 
89 

90 
91 
92 
93 
94 

95 
9i> 
97 
98 
99 

100 
101 
102 
103 
104 

105 
106 
107 
108 
109 

110 
111 
112 
113 
114 

115 
116 
117 
IIS 
119 


44.35  . 

44.47+°'^ 

44.59 

44.71 

44.83 

44.95 
45.07 
45.19 
45.31 
45.44 

45.56 
45.  CS 
45.80 
45.92 
40.04 

46.16 
46.29 
46.41 
40.53 
4U.65 

46.77 

46.,S9' 

47.01 

47.13 

47.24 


0.12 
0. 12 
0.12 
0. 12 

-1-0.12 
O.  12 
O.  12 
O.I3 
0. 12 

-t-0.12 
0.12 
O.  12 
O.  12 
O.I2 

+  °-M 
O.  12 
O.  12 
O.  12 
O.  12 


,+O.I2 
O.  12 
O.  12 
O.  II 
O.I2 

4 7.. 16  , 
47.48+°'^ 
47.5!) 
47.71 

47.82 


o.  I  I 
O.  12 
O.  II 
O.I  2 

47.94  , 

48.05+°-" 

48.16 

48.27 

48.38 


o.  I  I 
o.  I  I 
O.I  I 
O.ll 

48.49  , 

48.  ,59+°'° 
4,S.70  °" 

48.50  °'° 

48.91   °" 
o.io 

49.01  , 

49.11+°'° 

49.21   °'° 

49.31  °'° 

O.  ID 

49.50  J „ 

49.59+°-°'' 

49.77  °?'^ 

*'J«5  .In 
0.09 

49.94  ,„„o 

•'•»"   c"oS 

f-'-'-'fi  col 
fiO.34  , 
50.41 +°-°7 
f.0.48  °-°7 
50.,55  °°7 
50.62 


222.07  , 
223.30  + 
224.52 
225.73 
2a,:..  94 

228.14 
229.33  + 
230.51 
231.69 
232.86 

234.02  , 

235.17"^ 

236.31 

237.45 

238.53 

2.39.70  , 

240.81''" 

241.91 

243.00 

244.09 

245.17  , 
246.24"^ 
247.. 30 
248. ;{5 
249.38 

250.41 
251.43 
252.44 

253.44 
254.43 


+ 


.14 
•14 
•'3 

.11 
.  10 
.09 
.09 
.08 

.07 
.06 
■°5 
•03 

■°3 

.02 
.01 
.00 


0.99 
0.98 

S:^i+°-''r 

257.34  °-'^^ 
2.^8.29  °-'^5 
259.23  °-'^-' 


260.16 
261.08 
261.99 
262.88 
263.76 


4.0.92 
0.91 
0.N9 
0..S8 
0.87 

^65l9+°-«'^ 
S'l"-"'*  o.«4 
268.00 


0.81 


268.  SI 
269.61 
270.41 
271.19 
271.96 


120       50.69 


0.07 
0.07 


+0.S0 
o.Xo 
0.78 
0-77 
°-7S 

272.71   , 

273.45+°-7-» 

274.18 

274.90 

275,60         , 
0.09 

276,29  ,      .„ 

276.97+°'^'^ 

277.63 

278.28 

278.!»2 


0-73 
0.72 
0,70 


279,55 


o.M) 
0,^15 
o.().\ 
0.63 


1.78 
1.76 
1.75 
1.73 
1.72 

1.70 

1.69 
1.67 
1.66 
1.64 

1.63 
1.62 
1.60 
1.59 
1.57 

1.56 
1.55 
1.53 
1.52 
1.50 

1.49 
1.18 
1.46 
1.45 
1.43 

1.42 

1.40 
1.39 
1.38 
1.36 

1.35 
1.33 
1.32 
1.30 
1.29 

1.28 
1.26 
1.25 
1.23 
1.22 

1.20 
1.19 
1.17 
1.16 
1.14 

1.13 
1,11 
1,10 
1,09 
1.07 

1.06 
1,05 
1.03 
1.02 
1.00 

0.99 
0,98 
0,97 
0.96 
0,94 


270.69 

269.89" 

269.09 

268.28 

267.45 

266.61 

265.76" 

264.89 

264.02 

263.13 

262.23 

261.32" 

260.39 

259.46 

258.52 

257. .56 

256.59- 

255.61 

254.62 

253.62 

252.61 

251.59- 

250.56 

249.51 

248.45 

247.38 

246.30" 

245.21 

2.44.12 

243.01 

241.89 

210.76- 

239.62 

238,48 
237.33 

236.16 

234.98" 

233.79 

232.60 

231.39 

2.30.18 

228.96- 

227.73 

226.50 

225.25 

223.90 
222.73" 
221.46 
220.18 

218.88 

217.58 

216.28" 

214.97 

213.65 

212.33 

210.99 

209.65" 

208.31 

206.96 

205.60 


-0.80 
0.80 
0.81 
0.83 
0.S4 

-0.85 
0.87 
0.87 
0.89 
0.90 

-o  91 
0-93 
0-93 
0.94 
0.96 

-0.97 
0.98 
0.99 
00 


0,93     204,23 


0.14 
0.13 
0.13 
0.13 
0.13 

0.12 
0.13 
0.12 
0  12 
0.11 

0.11 
0.11 
0.11 
0.11 
0.11 

0.11 
0.11 
0.11 
0.11 
0.11 

0.11 
0.11 
0.11 
0.11 
0.11 

0.11 
0.12 
0.12 
0.12 
0.12 

0.12 
0.12 
0.12 
0.13 
0.13 

0.13 
0,13 
0.13 
0.13 
0.14 

0.14 
0,1  t 
0.15 
0.15 
0.16 

0.16 
0.16 
0,17 
0,17 
0,18 

0,18 
0,1s 
(,19 
0,19 
0.20 

0.20 
0,21 
0,21 

0,22 
0.22 


238 
239 
240 
240 
241 

241, 
242, 
243, 
243. 
244. 

244. 

245. 
245. 
246. 
246. 

247. 
247. 
248. 
248. 
249. 

249. 
249. 
250. 
250. 
251. 

251. 
251. 
252. 
252. 
252. 

252. 
253. 
253. 
1253. 
253. 

254. 
254, 
254. 
254. 
254, 

255, 
255, 
255. 
25.'-), 


^g+0.64 
12     °-^5 

'.     0.0 1 
0.60 

53+°-58 
10     °S7 

"T     til 

.y.2    °-55 

0.54 

79     °-5' 
0.50 

0.49 

0.48 

2fi  , 

7.) +0.47 

18     "--tS 
(;.,     9-44 

05     °-»-5 
0.42 

JJ7+0.40 

26 
64 


29 

78 


0.38 


01 


°-37 


37 
7[+o-34 

04     °-'53 

36     °-^' 

67     °--^' 
0.29 

50 

''     0.23 
0.23 


98 


0.26 
0.2; 


21 

^^«+°-; 

63 

82 
99 


o,  19 
0.17 
o.  16 


255, 
255, 
255. 
255. 

256, 
256. 
256, 
256. 
256. 

255, 
255, 
255, 
255, 
255, 


0.13 

O.  12 
O.  I! 
C.09 

^.,-l-o.oS 

90      °°7 

95     °-°S 

<)<)     004 

0.03 

02  , 

04+°°' 

04     °°° 

03-°-°' 

00     °°| 
0.04 

96 


'^4 
76 
67 

0,23    1255,56 


I 


0,07 
o.oS 
0.09 
0.1 1 


0.91 
0.90 
0.98 
0.87 
0.86 

0.84 
0.83 
0.82 
0.81 
0.79 

0.73 
0.77 
0.75 
0.74 
0.73 

0.71 
0.70 
0.69 
0.68 
0.66 

0.65 
0.64 
0.63 
0.62 
0.61 

0.60 
0.58 
0.57 
0.56 
0,55 

0,54 
0,53 
0,52 
0.51 
0.50 

0,49 
0,48 
0,47 
0,46 
0.45 

0,44 
0,43 
0.42 
0.41 
0,40 

0,39 
0,39 
0,38 
0.37 
0.36 

0,35 
0,34 
0,33 
0.33 
0.32 

0.31 
0.311 
0,29 
0,28 


'  r.e.2)  DilT.  Scc.var, 


192.67 

191.55' 

190.42 

189.29 

188.14 

186.99 

185.83' 

184.67 

183.51 

182.34 

181.16 

r....97" 

178.78 
177.59 
176.40 

175.20 
173.99- 
172.78 
171.. 56 
170.34 

169.11 

167.88" 

166.65 

165.41 

164.17 

162.92 

ici.rj- 

160.42 
159.16 
157.90 

156.63_ 

155..'!6~ 

154.09 

152.82 

151.55 

150,27 

I4s,!('.r 

147,71 

146,42 

145,13 

143.84 

142.55" 

141.26 

139.97 

13><.n7 

1 37.. 37. 

136,07" 

134.77 

133.47 

132.17 

130.87 
129,57" 

128,2(; 

1 21;.  96 
125,66 

121  36 

123,05" 

121,75 

120.45 

119,15 


12 

'3 

13 

»S 

'5 

16 

16 

16 

'7 

18 

»9 

«9 

•9 

>9 

20 

0  27      I17,s5 


-1. 21 
1. 21 
1.22 
1.22 
>-23 

-1.23 

1.24 
1.24 
'•-'5 


1.2O 
1.26 
1.27 

-1.27 
1.27 
1.27 
1.27 
1.28 

-1.28 
1.2.S 
1.29 
1.29 
1.29 

-1.29 
1.29 
1.29 
1.30 
1.30 

-1,30 
1.30 
1.30 
1.50 
1.30 

-1.30 

'•3> 
1,30 
1.30 
1.30 

-•■3' 
1.30 
1.30 
1.30 
'•30 


0.22 
0.23 
0.23 
0.24 
0.25 

0.25 
0.26 
0.27 
0.28 
0.28 

0.29 
0.30 
0.31 
0.32 
0.33 

0.34 
0.34 

0.35 
0.36 
0.37 

0..38 
0.39 
0.40 
0.41 
0,42 

0,43 
0,43 
0,44 

0.45 
0.46 

0.47 
0.48 
0.49 
0..50 
0.51 

0.52 
0.54 
0.55 
0.56 
0.57 

0.58 
0  59 
0.60 
0.61 
0. 1!2 

0.63 

0.'!5 
0.66 
0.67 
0.68 

0,69 
0,70 
0,71 
0.73 
0.74 

0.75 
0.76 
0.77 
0.78 
0.80 

0  81 


TUB   ORBIT  OP   URANUS. 


287 


T.\I5LE  IX, 

Abo.  2 

. — Continued. 

1 

Avg. 

(r.,s.3) 

(..C.3) 

(V.sA) 

((!.C.4> 

(p.e.O) 

(p.«.l) 

(p.C.l) 

(p.«.2) 

(p.c.2)  (p.8.3)  (h.c.3)| 

r,o 

If 
10.20 

It 

9.11 

n 

1.80 

It 

0.89 

1490 

437 

2222 

583 

127 

137 

71 

fii 

lo.i;9 

9.00 

1.79 

0.88 

1489 

445 

2233 

580 

125 

137 

70 

(12 

10.32 

8.88 

1.79 

0.87 

1483 

453 

2243 

577 

123 

137 

70 

(i:i 

10.35 

8.70 

1.78 

0.85 

1477 

400 

2254 

574 

121 

137 

70 

G4 

10.38 

8.04 

1.78 

0.84 

1470 

40S 

2204 

570 

119 

137 

69 

(15 

10.41 

8.52 

1.7S 

0.83 

1404 

470 

2275 

507 

117 

130 

69 

or, 

1G.43 

8.40 

1.77 

0.81 

1457 

484 

2285 

504 

115 

130 

68 

f.7 

10.45 

8.28 

1.77 

0.80 

1451 

492 

2295 

500 

113 

130 

68 

(IS 

10.47 

8.10 

1.70 

0.79 

1444 

501 

2300 

557 

112 

130 

67 

(19 

10.49 

8.04 

1.75 

0.78 

1438 

509 

2310 

553 

110 

130 

67 

70 

10.50 

7.91 

1.75 

0.70 

1431 

518 

2320 

550 

108 

130 

66 

71 

10.51 

7.79 

1.74 

0.75 

1424 

527 

2330 

640 

107 

130 

60 

72 

10.52 

7.07 

1.74 

0.74 

1417 

530 

2340 

543 

105 

130 

60 

73 

10.53 

7.55 

1.7! 

0.73 

1410 

545 

2350 

640 

103 

135 

65 

74 

10.54 

7.43 

1,72 

0.71 

1403 

554 

2305 

530 

102 

135 

65 

75 

10.54 

7.31 

1.72 

0.70 

1390 

603 

2375 

533 

100 

135 

04 

7(1 

10.54 

7.19 

1.71 

0.09 

1389 

572 

23S4 

630 

99 

135 

64 

77 

10.54 

7.07 

1.70 

0.08 

1382 

581 

2394 

520 

VJS 

135 

64 

78 

10.54 

0.95 

1.70 

0.07 

1374 

590 

2403 

523 

90 

135 

63 

79 

10.53 

0.83 

1.09 

0.05 

1307 

599 

2413 

519 

95 

135 

63 

80 

10.53 

0.71 

1.08 

0.04 

1.300 

008 

24  2f! 

510 

94 

135 

02 

81 

10.52 

0.00 

1.07 

0.03 

13.53 

018 

2431 

512 

93 

135 

02 

«2 

10.52 

0.48 

1.00 

0.02 

1340 

027 

2440 

608 

92 

134 

01 

83 

10.51 

0.30 

1.05 

0.01 

1339 

037 

2449 

505 

91 

134 

01 

84 

10.50 

0.24 

1.04 

0.00 

1332 

640 

2457 

501 

89 

134 

00 

85 

10. 4S 

0.13 

1.03 

0.59 

1325 

050 

2400 

498 

88 

134 

CO 

80 

10.40 

0.01 

1.02 

0,5S 

1318 

004 

2475 

494 

87 

134 

59 

87 

10.44 

5.89 

1,01 

0.57 

1311 

070 

24  S3 

490 

80 

134 

59 

88 

10.42 

5.78 

1,00 

0.50 

1304 

084 

2492 

4X7 

85 

134 

58 

89 

10.40 

5.00 

1.59 

0.55 

1290 

090 

2500 

483 

84 

133 

58 

00 

10.38 

5..55 

1.5S 

0.54 

1289 

700 

2509 

480 

84 

133 

57 

ill 

10.35 

5.44 

1.57 

0.53 

12SI 

717 

2517 

470 

83 

133 

57 

92 

10.33 

5.32 

1.50 

0,52 

1273 

727 

2525 

472 

82 

133 

50 

93 

10.30 

5.21 

1,55 

0,51 

1 200 

73S 

25.34 

409 

82 

132 

50 

94 

10.27 

5.09 

1.51 

0.50 

1258 

749 

2542 

405 

81 

132 

55 

!I5 

10.21 

4. 98 

1.5:t 

0,50 

12.50 

700 

2550 

401 

81 

132 

55 

ilCi 

10.20 

4.87 

1.52 

0,49 

1242 

771 

255  S 

457 

80 

131 

64 

!I7 

10.17 

4.70 

1,51 

0,4S 

1234 

782 

2500 

454 

80 

131 

54 

9S 

10.13 

4.04 

1,50 

0.48 

1220 

793 

2574 

45(1 

80 

131 

63 

99 

10.09 

4.54 

1.49 

0.47 

121S 

804 

25M2 

440 

79 

130 

53 

100 

10.05 

4.44 

1.48 

0.40 

1210 

815 

2590 

442 

79 

130 

52 

KiL 

10.01 

4.33 

1,40 

0.40 

1202 

S20 

259X 

43S 

79 

130 

51 

102 

15.90 

4.23 

1.45 

0.45 

1194 

83S 

2005 

435 

79 

129 

51 

103 

15.92 

4.13 

1,41 

0.44 

1180 

H49 

2013 

431 

78 

129 

5(( 

104 

15.87 

4.02 

1.43 

0.44 

1178 

801 

2020 

42S 

78 

129 

60 

105 

15.83 

3,92 

1.42 

0.43 

1170 

872 

2027 

424 

78 

128 

49 

100 

15.78 

3.82 

1.41 

0,43 

1102 

8><3 

2034 

420 

7s 

128 

49 

107 

15.73 

3,72 

l..i9 

0.42 

1154 

895 

2041 

410 

78 

128 

48 

108 

15.08 

3.02 

1.3S 

0,42 

'140 

907 

20  47 

413 

78 

127 

4S 

109 

15.02 

3.52 

1.37 

0,41 

1138 

919 

2054 

409 

78 

127 

47 

110 

15,50 

3.43 

1.30 

0,41 

1130 

931 

2001 

405 

78 

120 

40 

111 

15.51 

3,34 

1,.!4 

0,  10 

1122 

943 

2(;os 

402 

78 

120 

40 

112 

15.45 

3,24 

1,33 

0.10 

lilt 

955 

2074 

39S 

79 

125 

45 

113 

15.39 

3.15 

1,32 

0.  !(l 

11(10 

907 

20><1 

394 

125 

45 

114 

15.33 

3,(((> 

1,31 

0.39 

109S 

979 

20S7 

391 

79 

124 

44 

115 

15.27 

2.97 

1.29 

0,39 

1090 

991 

2093 

3S7 

80 

124 

44 

no 

15.21 

2.88 

l.-.'S 

0,3S 

10H2 

1003 

2099 

383 

SO 

123 

44 

117 

15.)  4 

2.  HO 

1,20 

0,:!8 

1074 

1010 

2705 

380 

80 

1 22 

43 

lis 

15.08 

2.71 

1,25 

0.3S 

1000 

1028 

2711 

370 

SI 

122 

43 

119 

15.01 

2.02 

1.24 

0.3S 

1059 

1(141 

2710 

373 

81 

121 

42 

120 

14.94 

2.54 

1.23 

0.37 

1051 

1053 

2722 

370 

82 

120 

42 

238 


THE   ORBIT   OP   URANUS. 


i 
I 


"II 


TAIJLK  IX,  . 

Vro.  2. — Continued. 

Ai'ff. 

(v.c.O)    Diff.  ,11.8.1)  Diff.  Soc.var. 

KV.C.l) 

Diff.  Sco.var. 

(V.g.2)  Dill'.  Scc'.var. 
''           'I          II 

(i'.c*.2)  Dilf.  Sccvar. 

11          II 

tf                         ft 

tt 

tf 

It 

/' 

//             tt 

II 

120 

60.09 

279.55  ,      , 
280.10+°-^' 

0.93 

204.23 

0.23 

255,50 

C.27 

117,85 

0.81 

121 

50.70 +°-°J 
50.82     °°^ 
50.88     °°^ 

0,92 

202. 8(!' 

->  37 
1.38 
1.38 

0.24 

255.44-°" 

0.20 

110.55-'-3° 

0.82 

122 

0.90 

201.48 

0.24 

2,55.31     °'^ 

0.20 

115,25     •-■'° 

0.84 

12:j 

281.35     °59 

0.89 

200.10 

0.25 

255.10     °-   S 
254.00     °'^' 

0.25 

113,95     '••5° 

0.85 

124 

50.93     °°5 

281.92     °-5J 

0.88 

198.71 

« ■  M) 

0.25 

0,25 

112,05     '--5° 

0,87 

0.05 

0.56 

1..10 

0.17 

1.29 

125 

50.98^^    - 

282.48  , 

0.87 

197.31 

0.20 

254.83 

0,24 

111.30 

0.88 

121! 

51.03 +°°3 

28,3. 02+°- 5* 

0.85 

195.91' 

-1.40 

0.27 

254.04-°-  ^ 

0,23 

110.07-!  -^5 

0.89 

127 

51.08     °-°-^ 

283  55     °-5.? 

0.84 

194.50 

1. 41 

0.27 

254,44     °-  ° 

0,23 

108.77       •^° 

0,91 

12H 

51.13     °°-^ 

2,84.07     °-5; 

0.83 

193.09 

1.41 

0.28 

2.54,23     "• 

0,22 

107,48     ]:l 

0.92 

129 

61.17     °°-* 
0.05 

284.57     °:^; 

0.81 

191.07 

1.42 
1.4J 

0.28 

254,00     l:ll 

0.22 

100,19   1:;^ 

0.94 

130 

51.22 

51.20 +°°-* 

285.00 

0.80 

190.25 

0.29 

25.3.70 

0.21 

104.91 

0.95 

i;ii 

285. 53+°- -'J 

0.79 

IS8.82' 

-'•43 

0.30 

253,51-°- -5 

0.21 

103.03-'-=^ 

0.96 

1.12 

51.30     °°-* 

285.99     °:'' 

0.7  < 

187.39 

«.4,^ 

0.31 

253.24     °-7 

0,20 

102.35     '-^^ 

0,98 

133 

51.34     °°-* 

280.44     °-'5 

0.70 

1S5.95 

1.44 

0.32 

252,90     °-'^ 

0.20 

101.07      '-^^ 

0,99 

134 

51.37     °°' 
0.03 

280.87     °-'' 
0.42 

0.75 

184.51 

1.44 
1.44 

o.;i3 

252,07     °'^ 
0,31 

0.19 

99.79     '-^^ 
1.27 

1.01 

135 

51.40 

287.29 

0.74 

183.07 

0.33 

252,30 

0.19 

98.52 

1.02 

130 

51. 43+°°^ 

287.09 +°-*° 

0.73 

181.02- 

-1-43 

0.34 

252,04 -°''^ 

0.18 

97.25 -'-7 

1.03 

137 

51.45     °°' 

2  •(8! 08     °-,59 

0.72 

180.17 

1-45 

0.35 

251.71     °-" 

0. 1 8 

95.99     '-^ 

1.05 

138 

61.47     °°-' 

288.45     °--'7 

0.70 

178.72 

1.45 

0.30 

251.30     °-35 

0.17 

94.73     '•-^' 

l.OG 

139 

51.49     °°- 
0.02 

288.81     °--5'' 
°j5 

0.09 

177.26 

1.46 
1.46 

0  37 

251.00     °i^ 
0-37 

0.17 

93.47     '--^ 
1.26 

1.08 

140 

61.51 

289.10 

0.08 

175.80 

0.38 

250.03 

O.lfi 

92.21 

1.09 

141 

61.52+°°' 

289.49+°.'-' 

0.07 

174.33- 

-1.47 

0.39 

250.24-°- 39 

0.10 

90.90— '--5 

1.10 

142 

51.53     '^•°' 

289.80     °->' 

0.00 

172.80 

1.47 

0.10 

249.84     °-»° 

0.15 

89.71      '-^5 

1.12 

143 

51.54     °°' 

290.10     °-5° 

0.05 

171.39 

1.47 

0.41 

249.43     °-»' 

0.15 

88.40      '--5 

1,13 

144 

61.65     °°' 

0.00 

290.39     °-'> 
0.27 

0.03 

109.92 

1.47 
1.48 

0.42 

249.01     °-^' 
0.44 

0.14 

87.22     '-'•* 
1.24 

1.15 

145 

51.55 

200.00 
290.92 +°--'' 

0.02 

108.44 

0.42 

2-««-''''      „   ,r 

0.14 

85.98 

1.16 

140 

61.55     °°° 

0.01 

100.90- 

-!..(,'! 

0.43 

248.12-°-'»5 
J,.  ,,,,     0.46 

247.18     °-f 

0.14 

84.75-'-^3 

1.17 

147 

51.55     °°° 

291.10     °-^ 

0.00 

105.48 

1. 48 

0.44 

0.13 

83.52     '--'3 

1.19 

14< 

51.54-°°' 

2'.ll,38     °-" 

0.58 

104.00 

1.48 

0.45 

0.13 

K..  ...)      "-23 

1.20 

149 

51.53     °°" 
0.01 

291.59     °-' 
0.20 

0.57 

';;2.52 

1.48 
1.49 

0.40 

240.09     °-^'' 
0.50 

0.12 

81,07     '■'' 
1,22 

1.22 

150 

61.52 

201.79 

0.50 

101.03 

0.47 

-+'l'"     0.. 

0.12 

79.85 

1.23 

151 

61.50-°°^ 

291.97+°'° 

0.55 

159.54- 

-I.. 19 

0.!8 

245.08-°-5' 

0.12 

7.S.(i4-'--' 

1.24 

152 

51.49     °°' 

292.13     °"' 

0.51 

158.05 

1.49 

0.49 

21.5.15     °-5-3 

0. 1 2 

77,44     ••^° 

1.20 

153 

51.47     °°- 

292.28     °'-i 

0.53 

150.50 

1,49 

0.50 

244.01     °-5j 

0.11 

70.24     \l° 

1.27 

154 

51.45     °°- 

292.42     °'-* 

0.52 

155.07 

1.49 

0.51 

244.00     °-5-! 
0,50 

0.11 

75.04      '•^° 

1.29 

0.02 

0. 12 

I  49 

1. 19 

155 

51.43 

292.54 

0.51 

153.58 

0.52 

243,50 

O.U 

73.85 

72.(;7--  '^ 

71.49       •   '; 

1.30 

150 

51.40-°°' 

292.04+°'° 

0.50 

152.09" 

-1.49 

0.54 

242,92  — °-S" 

0.11 

1.31 

157 

51.37     °°' 

292.73     °°'^ 

0.49 

150.00 

1.49 

0..-.5 

242  .■(3     ° ■-■''> 

0.11 

1.33 

158 

51.34     °°'> 

292,  KG     °°7 

O.H 

149.11 

1.49 

0.50 

241,73     °f° 

0.10 

70.;!2       - 

1.34 

159 

61.30    °■°^ 

292.80     °°'' 

0.47 

147.02 

1.49 

0.57 

241,12     °'" 

0.10 

09.15     '-'J 
1.16 

1.30 

0.04 

0.04 

'5° 

0.63 

100 

51.20 

292.90 

0.40 

140.12 

0.58 

240.49 

0.10 

07.99 

l.:i7 

101 

61.22-°°» 

292.9.3+°°,' 

0.45 

144.03- 

-1.49 

0.59 

239.,S)-°-^'4 

0.10 

0(1. S4  — '-'5 

1.39 

102 

51.17     °°' 

292.91     °°' 

0.4t 

143.13 

1.50 

0.00 

239.20     °''S 

0.10 

05.09     '-'5 

1.40 

103 

61.12     °°' 

292.94     °°° 

0.  (3 

141.03 

1.50 

0.01 

238.54     °-^''^ 

0.10 

04.55     '-'4 

1.42 

104 

61.07     °°' 
0.06 

292.92"°°" 

0.42 

140.14 

1.49 

0.02 

237.87     °-^'7 

0,10 

0-.t.41      '•'■* 

1.43 

0.04 

'5° 

0.69 

'•'3 

105 

61.01 

5o.y5-°-°^' 

50.89     °°'^ 

292.88 

0.41 

138.04 

0.04 

237,18 

0.09 

02.28 

1.45 

loo 

292  m;)— °°.^ 

0.40 

137.15- 

-1.49 

0.05 

230. 4  8-°- 7° 

0.(19 

01. 10-''^ 

1.47 

107 

292^70    °■°^ 

0.40 

135.00 

1.49 

0.00 

2,!5.77     °-'' 

0.<I0 

(lO.O.'i     '•" 

1.48 

108 

50.82     °°J 
50.70     "°^' 

292.08     °°^ 

o.;i9 

134.17 

1.49 

0.07 

235.05     °"- 

0.09  ; 

5S.04      '•" 

1.50 

109 

292  58     °' '° 

0.38 

132.08 

1.49 

0.08 

231,32     °-7,'> 

0.09 

57.84     '•'° 

1.51 

0.07 

0. 1 1 

1.48 

0,74 

1.09 

170 

50.09 

292,47 

0,37 

131.20 

0.09 

233,58 
232..S2— °7'' 

0.09 

50,75 

1.53 

171 

60.02-"°7 

292., 'U  ~°-'-' 

0.30 

129.71- 

-1.49 

0.70 

0.09  ; 

.5.5.(;7-'°« 

1.54 

172 

.'.0.54     °'°^ 

2i|.)  10     °-  '5 

0.35 

IJS.23 

1.48 

0.71 

232.05     "■~l 

(1.(19 

54.59     '•"« 

1.50 

173 

50.40     °°'' 

2!»2!o3     °'^' 

0.:!5 

1  ■-'•;.  75 

1..18 

0.73    ! 

i>;j|    .>1        O.7.S 

0.10 

.53. -.2     '"^ 

1.57 

174 

50.37     °°'J 
0.09 

291,80     °'7 
0,19 

0.,i4 

125.27 

I.  ;« 
1.4S 

■    '■•    i 

23(».48     °i" 
0,80 

0,10  ; 

52.45     '•°7 
1,05 

1.59 

175 

50.28 

291.07 

0..33 

123.79 

0.7.^ 

229,08 

228.s7-°-^' 

228.05     °Z' 

227,22     °-^' 

220.37     °-'^P 
0.8O 

0.10    ' 

51.40 

1.00 

170 

.'.0.19-°°'' 

291.40-°-' 

0.32 

122.32- 

-1.47 

0.7<> 

0. 10 

50.,!5-'°-'5 

I.Ol 

177 

•-"••O     000 

291. 2»     °  ■'■ 

0.,!1 

120.85 

1.47 

0.77 

o.lo 

49.31      '"^ 

1.03 

17S 

•'•"•»!       n?o 

291,01      °^' 

0.,'!l 

119.38 

I. 17 

0.79 

0.11 

4.S28      '•°' 

1.04 

179 

^■•••«l  0°;:: 

290,70     °"'^ 
0,27 

0.,!0 

U.S. 91 

...17 
..46 

0.80 

0.11 

1 

47,20     '°-' 
1. 01 

1.00 

ISO 

4'.I.,S1 

290  49 

0,29 

110.45 

O.Sl 

225.51 

0.11     ' 

40.25 

1.07 

,T^£tf ^aiwN*'**— ~  -^  -. .     -T  - 


' 


THE   ORBIT   OP   URANUS. 


239 


TABLE  IX 

Alio.  2. — Continued. 

Arg. 

{V.S.3) 

(u.f.3) 

((>.«.  4) 

{u.cA)   (p.c.O) 

y^f.S.\){f.C.\) 

(p.«.2) 

(f.c.'l)\  (|,.».3) 

(p.c.3j 

120 

It 
14.04 

n 
2.54 

1.23 

0.37 

1 
1051 

1053 

2722 

370 

82 

120 

42 

121 

14. «7 

2.4(! 

1.22 

0.37 

1043 

1 0(;5 

2727 

300 

82 

120 

42 

122 

14.H0 

2.38 

1.20 

0.37 

1035 

1078 

2732 

302 

83 

120 

41 

12;$ 

14.73 

2.30 

1.19 

0.37 

1028 

1090 

2737 

359 

83 

119 

41 

124 

14. (iO 

2.22 

1.18 

0.37 

102U 

1103 

2742 

855 

84 

119 

40 

125 

14.59 

2.15 

1.17 

0.37 

1012 

1110 

2747 

352 

85 

119 

40 

12(5 

14.52 

2.07 

1.15 

0.37 

1004 

1129 

2751 

348 

85 

118 

39 

127 

14.44 

2.00 

1.14 

0.37 

99(i 

1141 

2750 

344 

80 

118 

39 

12S 

14.37 

1.93 

1.13 

0.37 

989 

1154 

2700 

341 

87 

117 

38 

12 'J 

14.29 

l.SC 

1.11 

0.37 

9;1 

1107 

2705 

337 

88 

117  • 

38 

l.!0 

14.21 

1.79 

1.10 

0,37 

973 

IHO 

2709 

334 

89 

110 

37 

l.il 

14.13 

1.72 

1.09 

0,37 

9(;5 

1193 

2773 

330 

90 

115 

37 

i;i2 

U.OC) 

l.C.O 

1.07 

0,37 

958 

1 200 

2777 

327 

91 

115 

30 

i;j3 

13.08 

1.59 

l.OCi 

0,37 

950 

1219 

2781 

324 

92 

114 

30 

134 

13.90 

1.53 

1.05 

0.38 

942 

1232 

2785 

321 

93 

113 

35 

l;i-) 

13. H2 

1.47 

1.04 

0.38 

035 

1245 

27S9 

318 

04 

113 

35 

l:i(i 

13.74 

1.41 

1.02 

0.38 

927 

1258 

2793 

314 

95 

112 

35 

l.J7 

13.(;(> 

l.:!5 

1.01 

0.38 

920 

1271 

2790 

311 

90 

111 

34 

l.JS 

13.57 

1.30 

1.00 

0.39 

012 

12S4 

2^i00 

308 

97 

110 

34 

131) 

13.49 

1.24 

0.99 

0.39 

905 

1298 

2803 

305 

99 

109 

33 

no 

13.40 

1.19 

0.97 

0.39 

897 

1311 

2800 

302 

100 

108 

33 

Ml 

13.32 

1.14 

O.liC, 

0.40 

889 

1325 

2S(I9 

298 

)(il 

108 

33 

142 

13.24 

1.09 

0.!15 

0.40 

882 

1338 

2.sn 

295 

103 

107 

32 

14:{ 

13.15 

1.04 

0.94 

0.  II 

875 

13.-2 

2X14 

292 

104 

100 

32 

144 

13.07 

0.99 

0.93 

0.41 

807 

1300 

2810 

289 

100 

105 

32 

145 

12.08 

0.94 

0.91 

0.41 

850 

1379 

2819 

280 

107 

105 

31 

1411 

12.90 

0.90 

0.90 

0.42 

852 

1393 

2821 

282 

109 

104 

31 

147 

12.81 

0.8f, 

0.X9 

0.42 

844 

1407 

2823 

279 

no 

103 

31 

14S 

12.73 

0.82 

O..SS 

0.43 

837 

1420 

2825 

270 

112 

102 

30 

149 

12.04 

0.78 

0.87 

0.44 

829 

1133 

2S20 

273 

113 

101 

30 

150 

12.55 

0.74 

O.SC, 

0.44 

822 

144  7 

2><28 

270 

115 

100 

30 

151 

12.40 

0.70 

0.S5 

0.45 

815 

1401 

2829 

207 

117 

99 

SO 

152 

12.38 

o.(;7 

0.84 

0.40 

807 

1474 

2831 

204 

118 

98 

29 

15.i 

12.20 

0.04 

0.83 

0.40 

800 

14SS 

2S32 

201 

120 

97 

29 

154 

12.20 

0.01 

0.82 

0.47 

793 

1502 

2X34 

258 

122 

90 

29 

155 

12,11 

0.58 

0.81 

0.48 

780 

1515 

2835 

250 

124 

95 

28 

15(1 

12.03 

0.55 

0.80 

0.49 

778 

1529 

28,3(1 

253 

120 

04 

28 

157 

11.94 

0.52 

0.79 

0.49 

771 

1543 

2X37 

250 

128 

93 

28 

15(S 

11. 8(; 

0.50 

0.78 

0.50 

704 

155(; 

2S37 

247 

129 

92 

28 

151) 

11.77 

0.47 

0.77 

0.51 

750 

1509 

2838 

244 

131 

91 

27 

ICO 

11.08 

0.45 

0.7(1 

0.52 

749 

r^s.'J 

2838 

242 

133 

90 

27 

ICl 

11.59 

0.43 

0.75 

0.52 

742 

1597 

2s;!H 

2.39 

135 

90 

27 

102 

11.51 

0.41 

0.74 

0.53 

735 

1011 

2838 

2.30 

137 

89 

27 

KI.J 

11.42 

0.40 

0.73 

0.54 

728 

1024 

2S37 

233 

139 

88 

27 

KU 

11.33 

0.38 

0.72 

0.55 

7:>1 

10.38 

2837 

230 

141 

87 

27 

1(!5 

11.24 

0.37 

0.7! 

0.5(5 

714 

1052 

2X30 

228 

143 

80 

27 

Kid 

11.10 

0.3(> 

0,71 

0.57 

707 

1005 

2x:!.-, 

225 

145 

85 

27 

1(!7 

11.07 

0.35 

0.70 

0.58 

700 

1079 

2x34 

222 

147 

84 

27 

i(;,s 

10.09 

0.34 

O.CiO 

0.50 

093 

1093 

2x.!3 

219 

149 

83 

27 

Kil) 

10.91 

0.3il 

0.C8 

0.00 

C80 

1700 

2833 

210 

JM 

82 

27 

170 

10.82 

0.32 

0.08 

o.(;i 

f.79 

1720 

2X31 

214 

153 

82 

27 

171 

10.74 

0.32 

0.07 

o.(;2 

072 

17.!4 

2829 

211 

150 

81 

27 

172 

lO.O'i 

0.3) 

O.CO 

0.03 

005 

1717 

2X28 

208 

158 

80 

27 

173 

10.57 

r  ■ 

O.C.C, 

0.04 

059 

1701 

2820 

20(1  ' 

1(10 

79 

27 

174 

10. 4!' 

..31 

0,05 

o.cs 

G52 

1775 

2S24 

204 

102 

78 

28 

175 

10.41 

0.31 

0.05 

0.00 

045 

1788 

2822 

202 

105 

77 

28 

I7(i 

10.3:» 

0.3! 

0.04 

0.07 

038 

1802 

2820 

199 

107 

70 

28 

177 

10  2-) 

0.31 

(..03 

0.08 

032 

1815 

::i7 

197 

170 

75 

28 

17S 

10.  li 

0.32 

0.03 

0.09 

025 

1 829 

2X15 

1 95 

173 

74 

28 

170 

10.00 

0.32 

0.03 

0.70 

018 

18J2 

2812 

192 

175 

73 

28 

180 

10.01 

0.33 

0.02 

0.71 

Oil 

1S55 

2810 

190 

178 

72 

28 

--CiJ 


d 


940 


THE   ORBIT   OF   U  11  ANUS. 


I  i     •■ 
i  I 


TAHLK  IX, 

Alio.  2.  —  Continued. 

\Tg. 

(y.c.O)    Diir. 

(w.s.l)  Diir.  Sfc.viir 

O'.C.I) 

Diir.  Sfc'.var. 

(0.H.2)  Dill'.  Sir.var 

(,i'.c,2)  Dill'.  Sce.var. 

//          II 

//                    ft 

/r 

It 

It 

II 

II           It 

II 

It          It 

ft 

18U 

49.81 

'^""•*!'     ^  ,„ 

0.29 

Ilfi.45 

-1.46 
1.46 

0.81 

225.51 
224.C4-°-^7 

0.11 

4C.25 

1.C7 

181 

49.71-°'° 

290.20-°-"' 

0.28 

114.99 

0.82 

0.11 

45.24-'°" 

1 .  (18 

182 

49.  CO     °]\ 

289.90     °-^° 

0.28 

113.53 

0.84 

223  77     °-**' 
222.89     °-"** 

0.11 

**-'^   ".Z 

1.70 

188 

49.49     °" 

289.59     °-^' 

0.27 

112.08 

>-4S 

0.M5 

0.12 

43.25     °'>'> 

1.71 

184 

49.37     III 

289. 2G        ^^ 
0'35 

0.20 

'  1I0.C3 

>-45 
«4S 

0.8C 

221.99     °-9° 
0.91 

0.12 

^^-^«  o::j; 

1.73 

185 

*^'^^       n   ,, 

2*^8.91     „    , 
288.55-°- 3^ 

o.2r, 

109.18 

0.88 

^-'-^"*      on. 

0.12 

i;;!!--? 

1.71 

18(1 

49.13-°''^ 

0.25 

107.74 

—'•44 

0.M9 

220.IC-°-''^ 

0.12 

1.75 

187 

49.00     °'^ 

0.24 

101;.  30 

1.44 

0.90 

219.23     °-"^3 

0.12 

39.40     °-'^5 

1.77 

188 

48.87     l\] 

287.78     111 

0.24 

104.87 

'■43 

0.91 

21s.  29     °-'^-* 

0.13 

3S.4C    °;l\ 

1.78 

189 

^«-^^  l:\l 

=^^^-"  ix. 

0.23 

103.44 

'•43 
1.42 

0.93 

217.34     °'^^ 
0.96 

0.13 

■'^■^■^  V,\ 

l.so 

I'JO 

^'^••'■l     „,. 

28C.95     „  ., 

0.22 

102.02 

0.94 

2'«-;^'^     „„, 

0.13 

•'"■'•'■'1       n  n. 

1.81 

191 

48.47-°-   j 

28C.05   °-;^^ 

0  21 

100.00  ■ 

—1.42 

0.95 

215.41 -°-9 

0.13 

•'!-'-7''~!oo 

1.82 

192 

48.33     °'   ;^ 

0.21 

99. 1 9 

1. 41 

1. 41 
1.40 
1.40 

0.97 

214.44     °'97 
213.4C     °-9« 

0.14 

•^^-^O     o!  9 
33.91       'Z 

1.S4 

19a 

^^•>»     o    1 

285.58     °-]l 

0.20 

97.78 

0.9S 

0.14 

1.S5 

194 

*«"^     lit 

0.20 

9C.38 

1.00 

212.4C  ;:°° 

0.14 

l.,S7 

195 

47.00 

2S4.no 

0.19 

94.98 

1.01 

211.45 

0.15 

■ji;;;--f 

l.ss 

19a 

47. 75-°- '5 

284.09—°--''' 

0.19 

93..59- 

-1-3') 

1.02 

210.43-'-°* 

0.15 

1.90 

197 

47. CO     °'5 
47.44     °'^ 

283..5fi     °-53 

0.18 

92.20 

'•.?9 
..38 

1.04 

209.40     '-"3 

0.15 

S;S  =t 

1.91 

198 

283.01     °-S5 
282.45     °-5'^ 

0.18 

90. .S2 

1.05 

20.S.37     '-°3 

0  15 

1  '13 

199 

47.28     °'f 

0.17 

89.44 

1.38 

1.07 

207.33     •-°-* 

O.IC 

28.77    °''^ 

1.94 

o.  i6 

0-57 

'•37 

'•OS 

0.S2 

200 

47.12 

281.88 

0.17 

88.07 

-1.36 

1.0s 

2(»C.28 

O.IC 

27.95         „ 

1.9c 

201 

4(!.95-°'7 

2«tl.29-°-S9 
2S().C8     °-^' 
280.0C     °-^' 
279.43     °p 

0.17 

8C.7r 

1.09 

205.22-'-°'^ 

0.17 

27.14-°-^' 

1.97 

202 

4C.7S     °-'7 

0.1c 

85.  ;m 

'  .55 

1.11 

204.15     ^■°l 

0.17 

2C.34     °'^° 

1.99 

20:i 

4i;.Cl     °'7 

0. 1 C 

84.01 

'  3,S 

1.12 

203.07     '-°'^ 

0.1s 

2-V55     °'7'> 

2.00 

204 

4C.44     °'7 

0.1c 

82.  CC 

'•.?5 

1.14 

201.98     '■°'> 

0.18 

24.78     °-77 

2.U1 

O.I  8 

0.65 

«  34 

1.09 

0.77 

205 

4C. 2fi 

27'*. 78        .. 
'>7s  12-°-'''' 
^77:44     °f 

0.15 

81.32 

1.15 

200.  S9 

0.19 

24.01 

2.03 

20(i 

4C.08-°;« 

0.15 

79.99" 

-'•.33 

I.IC 

199.79-'-'° 

O.-JO 

2.'t.2C-°'" 

2.04 

207 

45.90     °'^^ 

0.15 

7H.C7 

'3^ 

LIS 

19S.C8      '-" 

0.20 

22.52       ''■* 

2.05 

201 

4.^.71     °"^ 

27r,.75     °-'^9 

0.14 

77.3C 

'3' 

1.19 

197. 5C      '-'^ 

0.21 

21.79     °'!-' 

2.  OC 

209 

45.52     °"^ 
0.19 

27C04     °-7' 
0.72 

0.14 

70.05 

'3' 
1.30 

1.21 

19C.43      ■-'•' 
'•'3 

0.21 

^'-0^    0:;: 

2.08 

210 

45..'53     ^  , 

275.32 

O.M 

74.75 

1.22 

195.30 

0.22 

20  3C     „  , 

2.09 

211 

4.V14-°"' 

274..58-°-^^ 

0.13 

73.40" 

-1.29 

1.28 

1.23 

194. IC-'-'-* 

0.23 

19.CC-°'/° 

2.10 

212 

4i.94     °" 

273.83     °-7j 

0.13 

72,18 

1.25 

193.01      '-'5 
191.85      '-'^ 
190.  CO     '-'" 

0.23 

IS  97      °'  "^ 
17. C4     .  , 

2.12 

2I:( 

4474     III 

273.0C     °!I 

0.13 

70.91 

1.27 

1.2c 

0.24 

2.13 

2U 

44.54     °"  272.28     "•'" 

0.13 

r.'i  i;4 

1.27 

1.28 

0.25 

2.14 

0.20                   0.79 

1.27 

1. 17 

'           0.64 

21.-) 

44,34               271  49        „ 

0.12 

n^  ,; 

1.29 

189.52           „ 

0.2c 

17.00        f 

2.1c 

21  li 

44.13-°-^'    270.CS-°-^' 
4.3.92     °'!   2'19.SC     °-^^ 
4!. 71     °-"   2-.9.03     °'^'' 

0.12 

C7.12 

-1.25 

1.31 

18S.34  -'-'^ 

0.27 

15  13        , 

2.17 

217 

0.12 

fi5.S8 

1.24 

1..-S2 

1S7.15     \\l 

0.28 

2.1s 

21 S 

0.12 

C4.C5 

'•23 

1.34 

1S5.9C      '-'^ 

0.28 

2.19 

219 

43.50     III  2C8.18     °;«| 

0.12 

C3.42 

'-23 
1.22 

1.35 

184. 7C      '•'° 
1.20 

0.29 

^•r.3   -° 

2.21 

320 

43.28        ,     2fi7..32 

0.11   ' 

fi2.20 

1.37 

I83.5C      .  _ 

0.29 

1^05     „  ,, 

2  2" 

821 

43.0c -°"  2CC.44-°'^''' 

0.11 

CO.  99 

-1. 21 

1.38 

1S2.35-- 

0.30 

IV'r°: 

1  v>   xO              -' 

2.23 

222 

42.84     °ll  2C5.55     °>^'> 

0.11 

59.  XO 

1. 19 

1.40   ' 

isi.u     -; 

0.31 

2  25 

223 

42.C2     °"  2'U.C5     °''° 

on 

5^.(11 

1.19 
i.>8 

1.11  : 

179.92       - 

0.32 

12:27  °-5-l 

2.2c 

224 

42.40     °"  2C3.73     °  ^^ 

0.11 

57.43 

1.43  1 

17.S.C9     '■'•' 

0.33 

11.73  °-:| 

2.27 

0.23                   0.93 

1 

1. 17 

'•23 

0.52 

325 

42.17               2C2.S0 

0.11    ; 

5C.2C 

-1.16 

1.14 

177.4C 

0.34 

11.21 

2.29 

32(! 

41.94-°-*'  2'!1.85-°''.f 

0.11 

.55.10- 

1.45 

17  c.  22  •"'•*-♦ 

0.34 

10.70-°-^' 

2..-i0 

327 

41.71     °-^'  2C0.M9     °'>^ 

Oil 

53. 9C 

1.14 

K47 

174.97     '--5 

0.35 

10.20     °-?° 

2.31 

328 

41.48     °-^.? 

259  92     °^' 

Oil 

52.82 

1. 14 

1.48 

173.72     '-*5 
172.4C     '-"'5 

0  3C 

9.72     °-»'^ 

2.32 

329 

41.24     °'-* 

2.58:91     "-'^'^ 

0.11 

51.C9 

'•'3 

1.50 

0.37 

9.25     °-»J 

2.34 

1 

0.24 

1. 00 

1.12 

'•27 

0.46 

230 

41.00 

257.94 

0.11 

.50.57 

1.51 

171.19 

0.3s 

8.79 

2.35 

2:tl 

40.7C-°-^-+  25'i.03-'°' 

0.11 

49.4C- 

-I.I  I 

1.52 

1r,9.92-'--7 

0.39  1 

S.35-°-*-» 

2.3I-, 

•2:ii 

40.52     °='-*  25.5.92      '  °' 

0.11 

4S.37 

1.09 

1.54 

1CS.C5      '--• 

0.40 

1  ^2     °--»3 

2.37 

2;i:l 

40.28     °-t   251.89      ■■°' 

0.11 

47.28 

1.09 
1.08 

I.5.'i 

IC7.37      '•*'„ 

0.41 

7:50     °"»^ 

2.3S 

2.U 

40.03     °^-'   2.53.85     '"^ 

0.11    ! 

4C.20 

1.57     ICC.  09     '--'" 

0.42 

7.00     °-»' 

2. 39 

0.25                   1.06 

1 

1.07 

'29 

0-39 

2:55  , 

39  78                252.79 

0.11    ' 

45.13 

1.58     Ifil.SO 

0.12 

fi.Tf 

2.41 

23fi  ; 

39.53-°- '5 

251.72^ '"^ 

0.11 

44.08" 

-1.05 

l.co    ir,3  51 -'•''> 

0.43 

c..-t2-°--^;' 

2.42 

2:i7  1 

39.28     °-^^ 

250.  C4      '"'^ 

0.12 

43.05 

1.03 

L'!l      1C2.22     '^^ 

0.44 

5.9C     °--^'^ 

2.4.! 

2H  1 

39.03     °l^ 

249.55      '-""^ 

0.12 

42.02 

1.03 

1.03     ICO.  92     '■•''^ 

0.45 

5.  CI     °--« 

2.14 

2.9  1 

3S.77     °-^  248.45     j- 

0.12    ; 

41.00 

1 .02 

1^'54     ir.9.C2     '-■'° 

0.4C 

5.27     °--^-* 

2.45 

1.02 

'3' 

I 

°32 

240 

38.51                247.33 

0.12 

1 

39.98 

l^Ofi     158.31 

0.47 

4.95 

2.4C 

THE   ORBIT   OF   URANUS. 


241 


TABLE  IX 

Aua.  i.— Continued. 

Art,'. 

(i>.«.3) 

(C.C.3) 

(0.8.4) 

(w.f.4; 

(p.c.OJ 

(p.8.1, 

(p.f.l, 

(p.8.2 

>  (p.c.2 

(p.«..3; 

(p.c.3) 

180 

10.01 

tt 

0.33 

// 

0.02 

tt 

0.71 

Oil 

1855 

2810 

190 

178 

72 

28 

181 

9.93 

0.34 

0.03 

0.72 

004 

1869 

2807 

188 

180 

71 

28 

183 

9.85 

0.35 

0.61 

0.73 

598 

1882 

2804 

186 

183 

70 

28 

183 

9.78 

0.36 

0.01 

0.75 

591 

1895 

2801 

184 

185 

69 

28 

184 

9.70 

0.37 

0.61 

0.76 

584 

1908 

2798 

182 

188 

68 

28 

185 

9.03 

0.38 

0.60 

0.77 

577 

1921 

2794 

180 

190 

07 

29 

186 

9.55 

0.39 

0.00 

0.78 

571 

1935 

2791 

178 

192 

60 

29 

187 

9.47 

0.41 

0.00 

0.79 

564 

1948 

2787 

176 

195 

05 

29 

188 

9.40 

0.43 

0.60 

0.80 

557 

1901 

2783 

174 

197 

04 

29 

189 

9.33 

0,44 

0.59 

0.81 

551 

1974 

2779 

172 

200 

03 

30 

190 

9.26 

0.45 

0.59 

0.83 

544 

1987 

2775 

171 

202 

62 

30 

191 

9.19 

0.47 

0.59 

0.84 

538 

2000 

2771 

109 

205 

01 

30 

193 

9.12 

0.50 

0.59 

0.85 

531 

2013 

2700 

107 

207 

(,0 

30 

193 

9.05 

0.,'-.2 

0.59 

0.86 

525 

2025 

2702 

105 

210 

59 

31 

194 

8.D8 

0.64 

0.59 

0.87 

519 

2038 

2757 

103 

212 

58 

31 

195 

8.91 

0.56 

0.59 

0.88 

512 

2051 

2752 

161 

215 

57 

32 

196 

8.84 

0.59 

0.59 

0.90 

500 

2004 

2747 

159 

218 

50 

32 

19T 

8.^8 

0.01 

0.59 

0.91 

600 

2076 

2742 

157 

220 

65 

32 

198 

8.71 

0.04 

0.59 

0.92 

494 

20>^9 

2737 

1 50 

223 

54 

33 

199 

8.65 

0.06 

0.59 

0.93 

488 

2101 

2731 

154 

225 

63 

33 

200 

8.59 

0.69 

0.59 

0.94 

482 

2113 

2726 

153 

228 

52 

34 

201 

8.53 

0.72 

0.59 

0,95 

470 

2120 

2720 

151 

231 

51 

34 

202 

8.47 

0.75 

0.59 

0.97 

470 

2138 

2714 

150 

233 

50 

35 

20:j 

8.41 

0.78 

0.59 

0.98 

404 

2160 

2708 

148 

23(! 

49 

35 

204 

8.36 

0.81 

0.00 

0.99 

459 

2102 

2702 

147 

239 

4S 

30 

205 

8.30 

0..S4 

0.60 

1.00 

453 

2174 

2695 

140 

242 

47 

30 

206 

S.25 

0.«T 

0.00 

l.OI 

447 

2186 

2089 

H4 

244 

47 

37 

207 

8.19 

0. 90 

0.01 

1.02 

441 

2198 

2082 

142 

247 

46 

37 

20S 

8.14 

0.93 

0.01 

1.03 

430 

2210 

2675 

III 

250 

45 

38 

209 

8.08 

0.90 

0.01 

1.04 

430 

2222 

20(;h 

140 

253 

44 

39 

210 

8.03 

1.00 

0.03 

1.0.-) 

425 

2234 

2061 

138 

250 

43 

39 

211 

7.98 

1.03 

0.02 

1. 00 

419 

2215 

2051 

137 

259 

43 

40 

212 

7.93 

1.06 

0.62 

1.07 

413 

2257 

2040 

135 

202 

42 

41 

2l;i 

7.88 

1.10 

0.03 

1.08 

408 

2268 

2039 

134 

205 

41 

42 

214 

7.84 

1.13 

0.03 

1.09 

402 

2280 

2031 

133 

208 

40 

43 

215 

7.79 

1.17 

o.ot 

1.10 

397 

2291 

2024 

132 

271 

39 

43 

2ir. 

7.74 

1.21 

0.04 

1.11 

392 

2302 

2616 

130 

274 

39 

44 

217 

7.70 

1.24 

0.05 

l.U> 

386 

2313 

2008 

129 

277 

38 

44 

218 

7.05 

1.28 

0.00 

1.13 

380 

2324 

2000 

128 

2S0 

37 

45 

219 

7.01 

1.32 

0.66 

1.14 

375 

2335 

2592 

127 

283 

30 

46 

220 

7.57 

1.35 

0.07 

1.15 

370 

2340 

2584 

120 

286 

35 

40 

221 

7.53 

1.39 

0.08 

1.10 

304 

2357 

2575 

125 

2S9 

35 

47 

222 

7.49 

1.43 

0.09 

1.17 

359 

2307 

2567 

125 

292 

34 

47 

22{ 

7.45 

1.46 

0.09 

1.18 

354 

2378 

2558 

124 

295 

33 

48 

224 

7.41 

1.50 

0.70 

1.19 

S49 

2389 

2550 

123 

298 

32 

49 

225 

7.38 

1.54  ! 

0.71 

1.20 

344 

2399 

2541 

122 

301 

32 

50 

22(1 

7.34 

1.58 

0.71 

1.20 

339 

2409  1 

2532 

121 

304 

31 

50 

227 

7.31 

1.02 

0.72 

1.21 

334 

2420  ; 

2523 

121 

307 

30 

51 

228 

7.28 

1.06 

0.73 

1.22 

329 

2430 

2513 

120 

310 

30 

52 

229 

7.25 

1.70 

0.74 

1.23 

324  1 

2440 

2504 

119 

313 

29 

53 

230 

7.22 

1.74 

0.75 

1.24 

819 

2450 

2494 

lis 

316 

28  1 

54 

231 

7.19 

1.78 

0.75 

1.24 

314 

21i>0  ■ 

2484 

lis 

318 

28  I 

54 

2:12 

7.17 

1.81 

0.76 

1.25 

sn 

2470  1 

2474 

117 

?21 

27 

55 

2:!:j  ; 

7.14 

1.85 

0.77 

1.25 

305 

2480  i 

2464 

110 

321 

27 

50 

234 

7.11 

1.89 

0.78 

1.26 

300  : 

2489  ! 

2454 

110 

327 

26  ] 

57 

235 

7.09 

1.93 

0.79 

1.27 

296 

2499  j 

2444 

115 

330 

25  '' 

58 

23fi 

7.07 

1.97 

0  80 

1.27 

291 

2508  ; 

2433 

114 

333 

25 

59 

237 

7.05 

2.00 

0.81 

1.28 

2«7 

2518  1 

2423 

114 

330 

24 

60 

238 

7.03 

2.04 

0.82 

1.28 

283 

2527  ■ 

2412 

113 

339 

23 

01 

239 

7.01 

2.08 

0.83 

1.29 

278 

2536 

2401 

113 

342 

22 

62 

240 

fi.99 

2.12 

0.84 

1.30 

274 

2545  ■ 

2390 

113 

345 

21 

53 

31  Jul7. 

1873 

^^^.m^t^ 

r 


2ii 


THE   OUBIT  OF   U  RAN  US. 


TABLE  IX,  Aho.  2 

. — Continued. 

Arg. 

(u.c.O)    Uiir.  j(u.s.l)  Diir.  S 

or.vor. 

(u.c.l)  Diir.  Soc.vor. 

(.U.S.  2)  1) 

IT.  8 

cc.vur. 

(V.C.I)  Diir.  S 

L'e.vor. 

tl                    H                         It 

/ 

ti 

ft                      H 

It 

It 

It 

It 

/'              /; 

// 

240 

38.51           f   247.33 
38.25-°-^^  24f..20-' 
37.99     °- 7  245.00     ' 

0.12 

39.98 

1.66 

158.31 

0.47 

4.95 

2.46 

241 

■13 

0.12 

38.98-'-°° 
3S.0O     °'^l 

1.67 

157.00-' 

•3' 

0.48 

*■'•■*  "o'lo 

2.47 

242 

'4 

0.12' 

1.09 

155.69     ' 

3' 

0.49 

4.34     °-° 

2.48 

243 

37.73     °-   ^ 

243.91 

'S 

.16 

18 

0.13 

■J7  ni    °97 

1.70 

154.37     ' 

•3* 

0.50 

4.00     °-    '^ 

2.49 

244 

3^-*«     III 

242.75     1 

0.13 

1.71 

153.05     ■ 

■i- 
•3» 

0.51 

3.79     °- 

0.25 

2.50 

245 

37.20    „  „ 

241.57 

.18 

0.13 

•''•''•12     „„, 

1.73 

151.73     , 

0.53 

3  •'•*     „,, 

2.51 

24G 

3r..93-°-^7 

240.39- 

0.13 

34.l9-°-W 

1.74 

150.40" 

33 

0.54 

3.;!0-°-  1 

2.52 

247 

SC.Ofl    °- 

239.20 

.19 

0.13 

33.27  °:i\ 

1.75 

149.07 

•33 

0.55 

30T     °-  f 

2.53 

243 

30.39     °- 

237.99     ' 

.  21 

0.14 

"2.36     °'5' 

1.76 

147.74 

■ii 

0.50 

2-^«     o'   0 

2.54 

249 

3«-ia   l\] 

236.77     j 

.  23 
.23 

0.14 

3l-*«     °oH 

1.78 

140.41     J 

■a 

•34 

0.57 

2"«     l\l 

2.55 

250 

35.85     .  ,, 

235.54     , 

0.14 

30.57     „  ,, 
29.70-°-^?, 

^;-??  °s 

^'•'^     0.S2 

1.79 

145.07     , 

0.58 

2^8     „,, 

2.56 

251 

3.-..58-°-'J 
35.31     l\l 

^^     oM 
3^-7^     0.28 

234.30" 

24 

■  n 

26 

.28 

0.14 

1.80 

143.73 

•34 
.34 
•34 
•34 
•35 

0.59 

2'l~o    ? 

2.57 

252 

233.05     1 

0.15 

1.S2 

142.39 

0.00 

2"''    0    i 

2.58 

253 

231.80 

0.15 

1.83 

141.05 

0.01 

2  02     °-   1 

2.59 

254 

2.50.54     1 

0.16 

1.85 

139.71     , 

0.02 

'■«»     TA 

2.60 

255 

.34.19-°*? 

229.20 

.28 

0.10 

2.5.52-°-^' 

1.80 

138.. 30 

0.64 

1.78 

2.61 

256 

227.98-' 

0.10 

1.88 

137.01-' 

•35 

0.05 

1.08-°-'° 

2.61 

257 

33  91     °-^l 
3.3.03     °-^^ 

226.08     ' 

30 

0.17 

24.73     °-" 
2.3.95     °ll 

1.89 

135.00     ' 

•35 

0.00 

1.00     °°^ 

2.62 

258 

225.37     ' 

3' 

0.17 

1.91 

134.31      ' 

•35 

0.07 

l.,53     °-°/ 
1.47      °-°^ 

2.03 

259 

33.34     l^l 

224.06     ' 

•3« 

0.18 

23.18     °-" 

1.92 

132.90     \ 

•35 

0.68 

2.04 

0.28 

•3» 

0-75 

•35 

0.04 

200 

33.00 
32.78-°'^ 

222.74 

0.18 

22.'»3_„,^ 

1.94 

131.01 

0.09 

'■^3     „„, 

2.05 

2ni 

221.41-' 

■M 

0.18 

21.69-°-74 

1.95 

130.20"' 

•35 
•36 

0.70 

l-^O^oof 

2. 00 

2C2 

32.49     °\l 

220.07     ' 

.?4 

0.19 

20.97     °-^ 

1.97 

128.90     ' 

0.71 

1.397°-°' 

2.07 

2fi3 

32.21     °'^ 

218.72     ' 

•.?S 

0.19 

20.26     °-    ' 

1.98 

127.. 55     ' 

•35 

0.73 

'-^0+°° 

2.07 

2U4 

31.92     °-'9 
0.29 

217.36     ' 

-.0 
•37 

0.20 

»«-S«     0^8 

1.99 

120.20     : 

•35 
35 

0.74 

1  -"a  °° 
0.03 

2.08 

2r,5 

SI"'' 

215.99 

0.20 

18.88     „  , 
18.21-°-^' 

n.56   °-^. 
16.92   °;l\ 

2.01 

124.8.'> 

0.75 

'■4!^  ,„„- 

2.69 

2fir, 

3!..34-°-''' 

214.02—' 

■.?8 

0.21 

202 

123.50—' 

•35 

0.-0 

1..5u+°-°5 
,  -,.     0.06 

2.70 

2(;7 

31.05     °-"' 

213.24     ' 

0.21 

2.03 

122.14     ' 

0.77 

2.71 

2R8 

30.70     °-\l 

211.85     ' 

•39 

0.22 

2.04 

120.79 

•35 

0.T9 

'•"3     °'° 

2.71 

2U9 

30.47  °:\i 

210.45     ' 

.40 
40 

0.22 

2.0G 

119.44     \ 

■35 
•35 

O.HO 

1.72    ^-°^ 

'■°      O.IO 

2.72 

270 

^^  1"     «  ,„ 

209.05 

0.23 

'^■50     ol6 

2.07 

118.09 

O.SI 

1.82  . 

2.73 

271 

29.88-°-^° 

207.04-' 

41 

0.24 

2.  OH 

110.74-' 

•35 

0..X2 

'•'•'^"^o";! 

2.74 

272 

29.59     °-;2 

200.22     ' 

4^ 

0.24 

2.10 

115.39     ' 

•35 

0.S4 

2-07  0  i 

2.74 

273 

29.29     °-° 

204.79     ' 

43 

0.25 

'•'•"*     o'» 

2.11 

114.04      ' 

•35 

0..S5 

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THE   ORBIT   OF   URANUS. 


243 


TAIJI 

.K  IX, 

(».c.4) 

Ahu.  2 

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274 

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1.31 

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113 

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1.31 

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2571 

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112 

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19 

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1.31 

257 

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112 

356 

10 

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2587 

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1.32 

249 

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112 

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1.32 

242 

2611 

2299 

112 

308 

16 

72 

249 

6.84 

2.45 

0.94 

1.33 

238 

2018 

2287 

111 

371 

16 

73 

250 

6.83 

2.48 

0.95 

1.33 

234 

2020 

2275 

111 

374 

15 

74 

251 

0.82 

2.52 

0.90 

1.33 

230 

2033 

2203 

111 

377 

15 

75 

252 

6.82 

2.55 

0.97 

1.33 

227 

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380 

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1.33 

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80 

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1  34 

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112 

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6.78 

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1.34 

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112 

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180 

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1.33 

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114 

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1.32 

180 

2735 

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116 

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1.31 

106 

2761 

2004 

117 

430 

11 

98 

272 

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1.20 

1.31 

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117 

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273 

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1.21 

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100 

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159 

2770 

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119 

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107 

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1.28 

1.20 

144 

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244 


T  II  U   O  It  in  T   or   URANUS. 


TAIU.K  l\,  Aii.i,  2 

I'c.vur. 

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(K.O.O)    Dlff. 

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2.80 

344 

0.20 

90.80 

'■47 
1,46 

1.00 

2.80 

ao.«8     °.«0 

1.87 

47.03     '"i 
1.06 

2.80 

345 

10.24 

95. 4  r 

1  01 

14.40  , 

2,80 

20.80 

1.88 

48.09 

2.85 

3411 

10.05-°"' 

03.05" 

-'■45 

1.02 

15.05-t-°-59 

15.05     °^° 

10.20     °-^" 

10.89     °-^'» 
0.65 

8,81 

28.93— °-^7 

1.90 

4y.i5+'-°^ 

2.85 

347 

9.85     °'^° 

92.50 

'45 

1.04 

-Lx.\ 

28.08     "•p 

1.91 

.'>0.22     "  °7 

2.85 

34S 

y.fiii    °"> 

91.00 

'44 

1.05 

8.8-i 

27.2.1     °«S 
20.39     °-^-» 

1.93 

61.30     '"^ 

2.85 

34'J 

U.47     °"' 

89.03 

'43 

1.07 

8.82 

1.94 

52.40     '  '° 

2.84 

0.19 

1.42 

0.82 

1. 10 

350 

9.28 

88.21 

1.08 

17.54  ,      ., 
l8.20+°-^'^ 
18.88     °-?'* 
10.57     °-^"^ 

2.83 

25.57,        _ 

1.00 

53.50 

2.84 

35 1 

9,10-°'t 

80.70" 

-1,42 

1.00 

2.83 

i,^.,.:-o.8i 

1.97 

64.01 -t-'" 

2.83 

352 

8,02     °'^ 

85.38 

1,41 

1   11 

8.84 

83.97     °'^'> 

1.!I9 

55.73     '  '^ 

2.83 

353 

8.74     °'** 

83.98 

1,40 

1.12 

2.84 

83.18     °-79 

2.00 

50.80      '-'3 

2.82 

354 

8.57     °'7 
0.17 

82.58 

1.40 
'•39 

1.14 

20,28  °;;; 

8.84 

22.40     °78 
0.70 

2.01 

57.99     '■'•' 
«  '5 

2.88 

355 

8.40 

81.10 

1.15 

21  00j^„  ,. 

2.85 

21-«*     „„ 

2.03 

59.14 

2.81 

35(i 

8.23-°- 'J 
8.07     °'^' 
7.01      °'^ 
7.75     °-^ 

70.81" 

-1,3s 

1.10 

21. .4+0.   4 

2.85 

20.8!|— °"S 

2.04 

00.29 +'-'5 

2.80 

357 

78.44 

'37 

1.18 

22.49     °" 
23,2.5     °ll 

84.03     °r 
0.79 

2,85 

20.10     °" 

2.05 

f.1.45     '"5 

2.80 

35S 
35  !l 

77.07 
75.71 

'•37 
1.35 

'35 

1.19 
1.21 

8.. so 

19.43     °73 

18.71      °'^ 
0. 70 

2.00 
2.08 

02.02     ''7 

03.80    J  J;^ 

2.79 
2.79 

8(;o 

7.fiO 

74.30 

1.22 

34.82 

a.  80 

18.01 

2.09 

64. 9d 

2.78 

THE  Oil  HIT  OP   URANUS. 


24S 


TAULK  IX, 

A  HO,  S 

.^Cmtdnued. 

Arg. 

(».«,3) 

it 

8.92 

(U.C.8) 
3.35 

(«.f.4) 

n 

1.43 

{v.oA) 

{(..c.O 
110 

(p.".!) 

1581 

(h.«.2 
143 

(^..•.2 

(,,.»..  3) 

( p.t'.3 
133 

1.08 

2848 

512 

15 

;iui 

(1.92 

8.35 

1.43 

1.07 

1 09 

2S4!I 

1566 

144 

515 

15 

134 

!1(I2 

<!.93 

3.85 

1.44 

1.06 

108 

2849 

1.551 

146 

517 

15 

185 

:to;< 

(1.93 

8.34 

1.44 

1.05 

107 

2850 

15:i5 

147 

519 

16 

137 

ao' 

(1  93 

8.34 

1.45 

1.04 

106 

2850 

1620 

149 

622 

16 

138 

an.". 

(1.94 

8.84 

1.45 

1.03 

105 

2850 

l.M)5 

150 

624 

17 

139 

:ioii 

(1.94 

8.33 

1.46 

1.U1 

104 

28,50 

1490 

152 

626 

17 

140 

SUIT 

(1.94 

3.33 

1.46 

1.00 

103 

2850 

1474 

153 

628 

18 

141 

itds 

(!  '.!( 

8.32 

1.47 

0.99 

103 

28,50 

1459 

154 

631 

'.8 

142 

auo 

(J.  95 

8.31 

1.47 

0.98 

102 

2819 

1444 

166 

633 

19 

143 

a  10 

6.95 

3.31 

1.47 

0.96 

102 

2819 

1429 

157 

635 

20 

144 

an 

(i.95 

8.30 

1.47 

0.95 

101 

28(8 

1413 

158 

538 

20 

145 

ai-j 

(1.95 

3.29 

1.48 

0.94 

101 

2847 

1398 

160 

640 

21 

146 

aia 

6.05 

3.28 

1.4S 

0.93 

KM) 

2847 

1383 

162 

642 

22 

147 

au 

6.94 

8.27 

1.48 

0.92 

100 

281(1 

1368 

163 

645 

23 

148 

ai5 

6.94 

8.26 

1.48 

0.90 

9!) 

2815 

1352 

165 

547 

24 

149 

aifl 

6.94 

3.25 

1.48 

0.89 

99 

2843 

1337 

167 

649 

24 

1.50 

an 

6.94 

8.23 

1.48 

0.88 

99 

2842 

1322 

168 

651 

25 

151 

ais 

6.93 

3.22 

1.48 

0. 86 

98 

2810 

1307 

170 

6.54 

26 

152 

ai<j 

6.03 

3.21 

1.48 

0.85 

98 

2838 

1291 

171 

666 

27 

168 

320 

6.92 

3,20 

1.48 

0.84 

98 

2836 

1276 

173 

668 

27 

154 

a-.'! 

6.92 

3.18 

1.48 

0.83 

98 

2834 

1261 

175 

660 

28 

1.56 

ai'3 

6.91 

3.17 

1.48 

0.81 

98 

2831 

124(1 

176 

562 

29 

156 

a-ja 

6.91 

3.15 

1.48 

0.80 

98 

2828 

1231 

178 

564 

29 

157 

324 

6.90 

3.14 

1.48 

0.79 

99 

2825 

1216 

180 

666 

80 

168 

325 

6.89 

3.12 

1.48 

0.77 

99 

2822 

1202 

181 

668 

31 

159 

:i2(i 

6.HS 

8.11 

1.48 

0.76 

99 

2819 

1187 

183 

570 

32 

159 

327 

6.H7 

3.10 

1.47 

0.75 

100 

28 15 

1172 

185 

672 

33 

160 

328 

6.«6 

8.08 

1.47 

0.74 

100 

2812 

1157 

I8(i 

673 

33 

161 

32!l 

6.85 

3.06 

1.47 

0.72 

101 

2808 

1143 

188 

575 

34 

162 

330 

6.83 

3.04 

1.46 

0.71 

101 

2804 

1128 

100 

577 

35 

163 

331 

6.82 

8.03 

1.46 

0.70 

102 

2800 

1113 

102 

679 

86 

164 

332 

6.81 

8.01 

1.46 

0.69 

103 

2796 

1098 

193 

581 

87 

165 

333 

6.79 

2.99 

1.45 

0.67 

103 

2792 

108  4 

195 

583 

38 

165 

334 

6.77 

2.97 

1.45 

0.66 

104 

2788 

1069 

197 

585 

39 

166 

335 

6.75 

2.96 

1.44 

0.65 

105 

2784 

1054 

199 

587 

40 

167 

33fi 

6.74 

2.94 

1.44 

0.64 

106 

2779 

1039 

201 

589 

41 

168 

337 

6.72 

2.92 

1.43 

0.63 

107 

2775 

1025 

203 

591 

42 

169 

33S 

6.70 

2.90 

1.43 

0,61 

108 

2770 

1010 

204 

592 

43 

169 

339 

6.  OS 

2.88 

1.42 

0.(10 

109 

2765 

906 

206 

594 

44 

170 

340 

6.66 

2.87 

1.42 

0.59 

110 

2760 

081 

208 

696 

45 

171 

341 

6.64 

2.85 

1.41 

0.58 

HI 

2755 

967 

210 

698 

46 

171 

342 

6.62 

2.8a 

1.40 

0.57 

112 

2750 

953 

212 

.599 

47 

172 

343 

6.59 

2.81 

1.40 

0.56 

114 

2744 

938 

214 

601 

48 

173 

344 

6.57 

2.79 

1.39 

0.55 

115 

2739 

924 

217 

608 

49 

173 

345 

6., 55 

2.78 

1.38 

0.53 

116 

2733 

910 

219 

605 

60 

174 

340 

6.52 

2.76 

1.37 

0. 52 

117 

2727 

896 

221 

606 

61 

174 

347 

6.50 

2.74 

1.36 

0.51 

119 

2721 

882 

223 

608 

62 

175 

34H 

6.47 

2.72 

1.36 

0.50 

121 

2715 

868 

226 

610 

53 

176 

34!) 

6.44 

2.70 

1.35 

0.49 

122 

2709 

865 

228 

611 

54 

176 

350 

6.41 

2.69 

1.34 

0.48 

124 

2703 

841 

230 

613 

66 

177 

35 1 

6.37 

2.67 

1.33 

0.47 

126 

2696 

828 

232 

614 

67 

177 

352 

6.34 

2.65 

1.32 

0.46 

128 

2689 

814 

235 

616 

68 

178 

353 

6.31 

2.64 

1.31 

0.45 

130 

2682 

801 

237 

617 

69 

178 

354 

6.28 

2.62 

1.30 

0.44 

132 

2675 

788 

230 

619 

60 

178 

355 

6.24 

2.60 

1.29 

0.43 

134 

2668 

775 

241 

620 

61 

179 

35r. 

6.21 

2.59 

1.28 

0.42 

136 

2660 

762 

244 

621 

62 

179 

357 

6.18 

2.57 

1.26 

0.41 

139 

2653 

749 

246 

623 

63 

180 

35.S 

6.14 

2.56 

1.25 

0.40 

141 

2645 

736 

248 

624 

64 

180 

359 

6.10 

2.51 

1.24 

0.39 

144 

2637 

723 

250 

626 

65 

180 

3(10 

6.07 

2.53 

1.23 

0.38 

146 

2629 

710 

252 

627 

67 

181 

I'l 


346 


THE   ORIJIT  OP   UilANUS. 


m 


!    .i 


■■: 


TAHL 

U  IX,  Aim.  8 

(c.f.i)  iiiir.  s 

—  Co, 
'c.vur. 

tinued. 

Arg. 

(i-.o.O)    Diir. 

ft                     t» 

(iM,.l)  IHir.  Soo.vur. 

(ox. 2)  Ditr.  S 

•c.  var. 

(o.c.2)  Diff.  Scf.viir. 

//              n              II 

ff                      If 

fl 

If             n 

It 

t'                    tl 

/* 

afio 

''■''■O        r.    ,r 

■J4.3n 

1.22 

25.(13+°-^' 
27.29     °;\\ 

2.8r, 

17.33-°^* 

2,00 

*"'*"8j..    ■„ 

2.78 

3til 

7.45 -°-'5 

7:t.02-'J-* 

1.23 

2.8(1 

2,10 

(1(1.17+  ■" 

2.77 

S(i2 

7.30    °-  5 

7I.«!»     ■•«3 

1.25 

2.87 

1(1.(1(1     °f7 
1(1.00     °"^ 
15.35     °"5 

2,12 

(17.37       •  ° 

2.77 

3fi3 

^••■^     o    ^. 

70.37     11^ 

1.21! 

2.87 

2,13 

118.58        • 

2.7(1 

3U4 

7.01     °'^ 

(10.05     '^' 

1.28 

2.87 

2.15 

00.79 

2.75 

O.I4 

>  3° 

0.64 

1.22 

3«!5 

3(iri 

1.20 
1.31 

20.00  ,^„„ 

2.88 
2.88 

u:ol-°t 

13.48     °-^' 
12.88      °'° 

2  111 
2.17 

72.23^'  ;, 

2.74 
2.74 

3tiT 

«■«»       o      ■ 

1.32 

30.78     °''° 

2.88 

2.18 

73.4(1     :• 

2.73 

3C8 

(!.4,S     °'-^ 

1.34 

31.(11  y' 

2.88 

2. 1 0 

74.70     \\\ 

2.72 

36  9 

C.3(!     °'' 

(;2.(;,i    ''7 

1.35 

32.(11     °'>^ 

2.80 

12.20     °-5'^ 

2.21 

''••'•'^       .-.2^ 

2.71 

O.  12 

1.25 

0.93 

0-57 

310 

r>  24 

f.l.I'.S     .  ,, 

1.37 

:<•■'•''•»    1  „       r 

2.80 

11.72 

2.22 

2.71 

3-1 

fi.l2-°'' 

«"H--^1 

1.38 

34.40+°'^5 
35.45     °'>^ 

2.8<.) 

"■'^~o    5 

2!  23 

2.70 

372 

«.00     °'' 

5S.01           • 

1.40 

2.80 

10.(13     °-^ 

2.25 

V.KU       ! 

2.(10 

373 

5.8»     °" 

57.f.H          ^ 

1.41 

3(1.42     °'^7 
37.40     °:^« 

2. 8!( 

10.10     °-  ■ 

2.2(1 

^""   :; 

2.(18 

374 

^•"  :;;; 

5(1.4(1  ,:,„ 

1.43 

2.80 

0.50     °-5' 
0.50 

2.27 

2.(17 

37.") 

S.fiH 

55.2fi 

1.44 

38  40 

2.80 

0.00 

2.20 

83.5B  , 

2.  fid 

:t7(; 

5.57-°" 

54.07-''9 
52  .SO       -'^ 

1.45 

30.41+'°" 

2.80 

8.(10-°-l'' 

2.30 

84.85+'-^'> 

2.(1(1 

377 

5.47     °'° 

1.47 

40.43     '°^ 

2.8il 

8.12     °-»;^ 
7(,(l    °-»'' 

2,31 

8(1.14      '■''> 

2.(15 

37S 

5.:i,s    """J 

51.72     "7 

1.4S 

41.47      ■•°* 

2,8!) 

2.32 

87.43      '•-" 

2.(14 

37» 

6.30     °°''' 

50.57     ''5 

1.50 

42.52     '"S 

2,80 

7.21     °-«S 

2.34 

88.73     •■•'° 

2.(13 

o.oS 

1.15 

I.Ob 

0.43 

'•3' 

3S0 

5.22 

*'^*'^     ,  ., 

1.51 

43.68  , 

2.80 

f..78 

2.35 

00.04  , 

2.(12 

3H1 

5.14-°°'^ 

4S/.1)-''' 

1.52 

44.(15  +  '  °7 

2.80 

(1.3(1 -°-«^ 

2.3(1 

01.35+'-'" 

2.(11 

3S'2 

5.07     °°l 

47.17  ' ;; 

1.51 

45  74     '°y 

2.8<l 

5.0(1     °-*° 

2.37 

02.(17      '••'' 

2.  (10 

HH3 

^  ,,,j     O.oS 

^'''•<""'          ,, 

1.55 

4(1.84     '•'° 

2,80 

5.10     °-«? 

2.38 

03.00         •'" 

2.50 

384 

^■^,;^   O.oS 

44.0(i     '  '° 

1.57 

47.05     '  " 

2.80 

2.39 

05.31      '■•'- 

2.58 

0.07 

1.09 

1.12 

0.36 

t  i.i 

:iSf, 

4.H4 

4;?.H7 

1.5S 

40,07  , 

2.80 

4.83 

2.40 

Of.. (14  , 

2.57 

3S!i 

4.77-°°; 

41.74     '"^ 

1.(10 

50.20  +  '  '^ 

2.80 

^^^~o-H 

2.42 

07.07 +'■•'■' 

2.57 

3S7 

4.71  ^■°': 

1.(11 

51.31      '   ';» 

2.8S 

4,15     °'- 

2.43 

0'.!.30      '••'•' 

2.5(1 

3SM 

4.(;5     °°^' 

4o.(;o      "S 

1.(13 

52.50     '  "^ 

2.SS 

3,8:i    ° 

2.44 

100.(14      '•■'* 

2.55 

38U 

4. CO     °°S 

30.  (!5     '°-» 

1  (14 

53.(17     '-'7 

2.88 

3,52     °-5' 

2.45 

101.08      '••'•* 

2.54 

0.04 

1.02 

1. 18 

O.il) 

'•35 

300 

4.-1(5 

3s.(i:i 

1.(1(1 

54.85 

2. 88 

S  -•■'     .  „ 

2.4(1 

103,33  j^,   ,. 

2,53 

H'.tl 

4.52-°  °-» 

:i7.(;2-'°' 

1.(17 

5(1.0»+'"> 

2.8,S 

2  0(1-° -J 

2  70    °;^ 

2.47 

101.(18 +'•■'■■' 

2,52 

3!l-2 

CO., 

:t5.(!.-,    °''l 

1.(10 

57.25     '  -' 

2.8K 

2.48 

10(1.03      '-'5 

2.51 

3'.);! 

4  45     °°-' 

1.70 

5s.4(l     '  -' 

2,87 

2  15     °'' 

2,40 

107.:i8     'll 

2,50 

3i)4 

4.41      °°-» 
0.0 ; 

••^■^•'     o..;6 

1.71 

50.(18      J  '1* 

2.87 

2,22     °-' 
0.21 

2,50 

2.40 

305 

4..'tS 

3:!  71 

1.73 

(1(1.01 

2,87 

2ni    „  ,„ 

2.51 

110.10  ,  ,      , 

2.47 

vm; 

4.:..-.-^^°-' 

32.77 -°''^ 

1.71 

(12.15  +  '  •'■♦ 

2.S7 

'■'*'"o     ^ 

2.52 

111.4.1+   •• 

2.4(1 

3!I7 

4.:t:J    °-^- 

31. S5     '^•'^ 

1.75 

(13,40      '  -5 

2,S7 

'•'•■=2    °^ 

2.53 

w>.^->     •' 

2.45 

3!iS 

4.31    °°- 

30.04     °'i 

1.7(1 

.14.(17       • 

.15.05  ;-« 

2.8C, 

l:^»  li 

2.54 

III.IS      [f 

2.44 

asiy 

4.:i0    °°; 
0.01 

="•■"3  :i; 

1.78 

2.8(1 

2.55 

1.5.55     \-H 

2.43 

4on 

4.20 

S;;i--i 

1.70 

(17.23  ,  .  , 

2.8r, 

l'«        n.. 

2.5(1 

'"■•"-.L.     „ 

2.42 

401 

4.2S-°°' 

1.80 

(ls..-,2+  -"> 

2.8(1 

l'"~o      • 

2.57 

lls,-20"^     •'' 

2.41 

40i> 

4.2s     °°° 

.-.42  °;'' 

1.S2 

(10,82        ••° 

2.85 

"'•"•       00 

2.58 

lio,.;(i    [ll 

2.40 

4ii;t 
4U4 

4.2,S     °°° 
4  2S     °-°° 

^5:7^     °:? 

1.83 
1.85 

71.13 
72.45     '•'' 

2.85 

2.84 

2.50 
2.(10 

121,03       •■" 
122,11         •', 

2.30 
2.38 

'•-4-0.01 

'        0.82 

1  .53 

0.07 

'■37 

4  Of) 

4  ''0 

2».01         0 

1.80 

73,78  . 

2.8» 

1        "■.■'■'' 

2.(10 

123.78  ,  ,     ^ 

2.311 

4  (It; 
4(17 
40S 
40!l 

:.^o+°°: 

2:!.3:t    °  !, 

1.88 
I.SO 
1.01 
1.02 

2,81 

2.8;| 
2.83 
2.82 

0.(10 -:;°5 

0.5.1      °°| 
0.00 

2.  Ill 
2.(12 
2.(13 
2.(14 

.25,l.l+- 
12(1,53       ■•" 

120,28         •" 
'■37 

2.35 
2.34 
2.33 
2.32 

410 

4.:iO 

21.  or, 

1.04 

8(1,50. 

2.82 

"•'••2l„„, 

2.(15 

1:10.(15 

2.31 

411 

4  4.) 4-00,^ 

20.;t3-°7.? 

1.05 

81,04 +  'V^ 

2,8-> 

0,51+°° 

2. '1(1 

l:i2.02  +  ''7 

2.30 

412 

4:4(1  °-°-' 

10.C.2     °J' 

1.07 

83.33      '    '■' 

2,81 

"•■'^      on- 

2.(17 

1:1:1,10     [■^>l 

2.20 

4l:l 

4..'-.0     °°-» 

IS.ilH      °    "> 

lS-<5     °'''* 
0.O7 

l.os 

84.73      '   '° 

2.81 

0.(12     °°S 

0.(18     °'°^' 
o.oH 

2.(17 

1:11.77     '••" 

2.27 

414 

4.54     °°' 

o.oi; 

1.00 

8(1.14      '    '" 
1.41 

2.80 

2.(l.s 

1:11;,  It     '-'^ 
'37 

2.20 

415 

4.50 

1    17. 5S 

2.01 

87.55 

2.80 

"f'''j_o„> 

2.(10 

1:17  51  , 

2.25 

41(1 

4.(14+°°.^ 

lC,.03-°'^ 

2.02 

88  .17+''" 

'.'.70 

0,85+°°" 

2,;o 

l.!S,8s  +  ''37 

2.21 

417 

4.70     °°^' 

1C..30     °M 

15.(;7    °',' 

2.03 

00.30      '    '■' 

2  70 

0,0(1     °" 

2,VI 

110  25     '••" 

2.22 

4H 

4.7(!     °°f 

2.01 

01.8-2     '••■' 

2.78 

1,08      °'' 

2  71 

111,,..     '37 

2.21 

41'.) 

4.,S2     °°' 
0.06 

,   i:..o(:    °'^" 

2.011 

i    0.!.2(1      '  ■" 
I  45 

2.78 

1      1.22     °'' 
o'5 

2.72 

11 -.',00     '■■'/ 
'■3'> 

2  20 

420 

4.S8 

14  47 

2.0, 

04.7! 

2.77 

1.37 

2,7:1 

lll.:!5 

2.19 

.J^ 


^ 


THE   OR  BIT   OP   U  11  AN  US. 


isn 


TAULK  IX, 

Abo.  2. — Continued. 

AfR. 

(U.8.3) 

(U.C.3) 
2.53 

(i'.».4) 

(...r.4) 

(p.r.O)  (p.«.l) 

710 

fp.«.2) 
253 

(p.c.2) 
027 

(,..H.3) 

67 

(P.c.3) 
181 

SCO 

It 

f.07 

If 

1.23 

0.38 

140 

2029 

aiii 

0.02 

2.51 

1.22 

0.38 

148 

2021 

097 

254 

028 

08 

I8l 

.S(>2 

5.98 

2.50 

1.21 

0.37 

151 

2013 

085 

250 

629 

(.9 

182 

•MV.l 

5.94 

2.49 

1.20 

0.30 

153 

2005 

072 

259 

031 

70 

183 

■Mi 

5.90 

2.48 

1.19 

0.35 

156 

2590 

000 

201 

032 

71 

183 

3fi5 

5.85 

2.46 

1.17 

0.35 

!59 

2588 

047 

203 

033 

72 

183 

aiiti 

5.  Hi 

2.45 

1  10 

0.34 

102   2579 

035 

205 

034 

74 

183 

3  (17 

5.7r, 

2.44 

1.15 

0.34 

104   2570 

023 

207 

035 

75 

183 

3r)8 

5.72 

2.44 

1.14 

0.33 

107 

2502 

010 

270 

630 

70 

183 

3(i9 

5.C7 

2  43 

1.12 

0.32 

170 

2553 

698 

273 

038 

77 

184 

370 

5.fi3 

2.43 

1.11 

0.32 

173 

2544 

580 

274 

639 

78 

184 

371 

5.58 

2.41 

1  10 

0.31 

170 

25;i5 

.•>74 

270 

040 

79 

184 

37. 

5.53 

2.41 

1.08 

0.31 

180 

2520 

502 

279 

041 

80 

184 

373 

5.48 

2.40 

1.07 

0.30 

183 

2510 

550 

281 

042 

81 

184 

374 

5.43 

2.39 

1.05 

0.30 

180  ,  2507 

539 

283 

043 

h3 

184 

375 

6.38 

2.39 

1.04 

0.29 

190   2498 

528 

285 

645 

83 

181 

370 

5.33 

2.39 

1.03 

0.2!. 

194   2488 

517 

288 

040 

84 

184 

377 

5.28 

2.38 

I.OI 

0.29 

197   2479 

500 

290 

047 

85 

|8( 

3:» 

5  22 

2.38 

1.00 

0.28 

201 

2409 

495 

292 

048 

80 

184 

379 

5.17 

2.38 

0.98 

0.28 

205 

2400 

485 

294 

649 

87 

184 

380 

5.12 

2.38 

0.97 

0.28 

208 

24.50 

474 

297 

650 

88 

184 

3S1 

5.07 

2.38 

0.90 

0.28 

212 

2440 

403 

299 

051 

89 

184 

3H2 

5.01 

2.38 

0.94 

0.27 

210 

2429 

453 

302 

052 

90 

184 

3S3 

4.911 

2.39 

0.93 

0.27 

220 

2419 

443 

304 

053 

91 

184 

3S4 

4. '.to 

2.39 

0.91 

0.27 

224 

2408 

4.33 

300 

054 

92 

184 

3S-, 

4.84 

2.39 

0.90 

0.27 

228 

2397 

423 

309 

6.55 

94 

184 

3sr, 

4.79 

2. 40 

0.88 

.i..27 

232 

o--  .,  , 

413 

312 

650 

95 

184 

3S7 

4.73 

2.40 

0.80 

0.27 

230 

2374 

40i 

:  1  4 

0.57 

90 

184 

3.SS 

4.07 

2.41 

0.8'. 

0.27 

210 

230:! 

394 

317 

057 

97 

183 

3.S.» 

4. Ill 

2.42 

0.81 

0.27 

245 

2:!5! 

384 

319 

058 

98 

183 

3!t0 

4.55 

2.43 

0.83 

0.27 

249 

2310 

375 

322 

059 

99 

183 

3!ll 

4  49 

2.44 

0.81 

0.27 

253 

2328 

300 

321 

OOO 

100 

183 

■M-i 

4.43 

2.40 

0.79 

0. 27 

258 

2'ilO 

357 

327 

001 

102 

183 

3it:t 

4.37 

2.47 

0.78 

0.27 

202 

2.!05 

318 

329 

002 

103 

18a 

391 

4.31 

2.48 

0.76 

0.27 

200 

2293 

339 

332 

662 

104 

183 

39') 

4.21 

2.49 

0.75 

0.27 

271 

2281 

330 

334 

603 

105 

182 

:!9<i 

4.18 

2.51 

0.73 

0.28 

270 

2209 

322 

330 

661 

100 

183 

397 

4.12 

i.r.i 

0.72 

0.28 

280   2257 

313 

339 

665 

107 

182 

398 

4.00 

2.55 

0.70 

0.28 

«<5  !  2245 

304 

341 

060 

109 

181 

399 

4.00 

2.57 

0.09 

0.28 

29U 

2232 

296 

341 

007 

no 

181 

400 

3.93 

2.59 

0.07 

0.29 

294 

2220 

288 

340 

667 

111 

181 

401 

3.87 

2.112 

0.00 

0.29 

299 

2207 

280 

318 

608 

112 

IHI 

402 

3.. 80 

2.04 

0.05 

0.30 

304 

2195 

273 

351 

008 

113 

180 

403 

3.74 

2.00 

O.f.3 

0.30 

309 

2182 

205 

353 

609 

114 

!nO 

404 

3.08 

2.09 

0.0-i 

0.31 

314 

2170 

258 

355 

609 

115 

180 

405 

3.02 

2.72 

o.oo 

9.31 

320 

2157 

251 

358 

009 

110 

179 

400 

3. ^.5 

2  75 

0.59 

0.31 

325 

2144 

241 

301 

009 

no 

179 

407 

3.49 

2.78 

0.57 

0.32 

331 

2131 

237 

303 

670 

117 

179 

40S 

3.43 

2.81 

0.50 

0.33 

337 

2119 

230 

305 

670 

118 

178 

409 

3.31! 

2.84 

0.55 

0.33 

312 

2100 

221 

308 

670 

119 

178 

410 

3.30 

2.88 

0.53 

0.34 

34S 

2093 

217 

371 

670 

1 20 

178 

411 

3.21 

2. 92 

0.53 

0.35 

35  I 

2080 

211 

37." 

070 

121 

177 

412 

3.17 

2.95 

0.50 

O.30 

359 

2000 

205 

370 

670 

122 

177 

413 

3. 1 1 

2.99 

0.  19 

0.30 

305   2053 

199 

378 

671 

123 

170 

414 

3.04 

3.03 

0.48 

0.37 

•171   2030 

193 

381 

071 

123 

170 

415 

2.98 

3.07 

0,40 

0.38 

3. '7   2020 

187 

382 

671 

124 

175 

41f. 

2.92 

3.  II 

0.45 

0.38 

,183   2012 

isl 

38(1 

671 

125 

171 

417 

2.85 

3.15 

0.44 

0.39 

;.89 

I!I98 

17"! 

388 

671 

120 

174 

418 

2.79 

3.20 

0.42 

0.40 

39) 

1  9S5 

no 

391 

071 

120 

173 

419 

2. 73 

3  24 

0  (1 

0.41 

401 

1971 

1(,5 

393 

672 

127 

173 

420 

2.67 

B.29 

0.40 

0.42 

407 

195Y 

160 

390 

672 

128 

172 

248 


THE  ORBIT  OF   URANUS. 


i  •' 


TABLE  IX,  Alio.  2 

. — Continued. 

Arg. 

(u.c.O)    Diff. 

(i).8.1)  Diir.  Sccvor. 

(tj.c.l)  DiOf.  Secvar. 

(U.S. 2)  Diff.  Secvar. 

Cu.c.2)  Diff.  Scfvar. 

ff                  ft 

ft                   ft 

/f 

n 

/ 

II 

ti               n 

» 

>l              n 

If 

430 

<-88  ,„„, 

14.47 

2.07 

94.71  , 

2.77 

1.37  . 

2.73 

144.35  , 

2.19 

421 

*-"5+oo 

13.90-°" 

2.08 

90.10  + 

45 
46 

2.70 

l-'^"*"o'8 

2.74 

14.5.72+  -37 

2.18 

422 

so'-J    III 

0.03 

13.35     °-53 

2.10 

97.02     ' 

2.70 

2.74 

147,09       -37 
148.45       -36 

2.10 

423 

12.81     °-54 

2.11 

99.09 

47 

2.75 

l.!12     °'° 

2.75 

2.15 

424 

12.28     °S3 

2.13 

100.50     1 

47 

2.75 

ai3     °'' 

2.75 

149.80     '••'S 

2.13 

0.51 

48 

0.22 

>-35 

425 

5-2fi  ,„„„ 

11.77 

2.14 

102.04 

48 

2.74 

2.  •'?•">  ,  „     , 

2.70 

151.15  ,        , 

2.12 

4M 

5-a'^"^ooo 

ii.28--°-;^9 

10.80     °-»? 
10.34     If^ 

2.15 

103.52  +  ; 

2.73 

2.M+°-   ? 

2.77 

I52,50  +  '-3S 

2.11 

427 

5.44     °°'^ 

2.17 

105.01     J 

49 

2,73 

2«^     0 

2.77 

1.53.84     \\\ 

2.09 

428 

5-W     °?^ 

2.18 

100.50     J 

49 

2.72 

-'»     o's 

3.39  °: 

2.78 

155.10        35 

2.08 

429 

sea   °-  ° 

9.89     °-»S 

2.20 

108.00     : 

5° 

2.72 

2.78 

150.54     '-'5 

2.06 

o.io 

0-43 

5« 

0.30 

1-34 

430 

5. 73  ,„  ,„ 

9.40 

2.21 

109.51  , 

2.71 

3.09  , 

2.79 

157.88  , 

2.06 

431 

6.83 +°-° 

9.05 -°-»' 

2.22 

111.02+; 

5« 

2.70 

4.00+°-^ 

2.79 

150.22  +  j^; 

2.04 

432 

lit     -" 

8.00     °-]1 

'■''  oil 

i.h 

112..53     . 

5' 

2.09 

^■■''^     ot 

2.80 

100., 55      ,  •" 

2.02 

433 

2.24 

114.04     : 

5« 

2. 0-l 

4  07     °-'^-' 

2.80 

101.88        ••» 

103.20  i:^; 

2.01 

434 

2.25 

115.56     \ 

5^ 
53 

2.07 

2.81 

1.99 

435 

0.29 

IM 

2.27 

117.09  , 

2.00 

5.39  ,        . 

2.81 

104.52 

1.98 

43i; 

fi.41+O.I2 

T,-2:i—°-3i 

2.28 

118.62  +  ' 

53 

2.00 

.5.77 +°-^'" 

2.82 

105.84 +  ''3' 

1.91 

437 

fi.54     °'3 

c.h   °'J' 

2.29 

120.16     ' 

54 

2.05 

0.17     °-l° 

2.82 

107.15     '-31 

1.95 

438 

0.07     °^' 

0.02     °-3° 

2.30 

121.70     ' 

54 

2.04 

0.58     °->' 

2.83 

108.40      '-3' 

1.94 

439 

6.80     °'-5 
0.14 

6.34     °-^8 
0.27 

2.31 

123.24     ' 

54 
54 

2.03 

7.00     °-'' 
0.44 

2.83 

109.76     '-3° 
1.30 

1.92 

440 

0.94 

0.07 

2.32 

124.78  ,, 

2.02 

■J^*  .      .. 

2.84 

171.00 

1.91 

441 

7.08 +°'4 

5.H2-°='S 

2.33 

120.33  +  ' 

55 

2.01 

7  H!.+°-»5 

2.84 

172.35+'-'' 

1.00 

442 

7.22     °"l 

6.. 59     °'i 

2.34 

127.88 

55 

2,00 

8.84      °-*" 

2.85 

173.04     '•-'' 

1.88 

443 

7.37     °'5 

5.38     "-' 

2.30 

129.43     " 

55 

2.50 

2.85 

174.113     '-'2 

1.87 

444 

7.52    °'5 

5.18     °-° 

2.37 

130.98     ■ 

55 
56 

2.58 

9.33     °-»'' 

2.80 

170.21      '•^'* 

1.85 

0.15 

0.18 

o-5« 

1.27 

445 

8.31     °"^ 

5.00 

2.38 

132.54  , 

2.57 

0.84  ,      ^ 

2.80 

177.48 

1.84 

44r. 

4. 83-°- '7 

2.39 

I34.10  +  ' 

56 
56 
56 
56 

2.57 

10.  .30+°- 5' 

2.80 

17H.75  +  '-7 
180.01       '•^'' 

1.83 

447 

4.08     °'5 

2.40 

135.00     ' 

2.50 

10.8!)     °-5.i 

2.87 

1.81 

448 

4.55     °'3 

2.42 

137.22     ' 

2.55 

11.44     °-55 
12.00     °S'^ 

2.87 

181.20     '-'5 

1.80 

449 

4.43     °'^ 

2.43 

138.78     1 

2.54 

2.88 

182.51     '-^5 

1.78 

0.17 

0.10 

57 

0-57 

1.24 

450 

8.48  , 

4.M          - 

2.44 

140.35 

2.53 

12.57  ,      , 

2.88 

183.75 

1.77 

451 

8.05+°- "7 

4.25-°°? 

2,45 

lll.!)2  + 

57 

2,52 

i;uo+°" 

2.88 

184.1I0  +  ''-'' 

1.70 

452 

8.82     °\l 
9.18     °'» 

4.19     °°'^ 

2,40 

14.3.49     ' 

57 

2.51 

13.70     °  '° 
14.37     °-'' 

2.88 

180,:!2     '-3 

1.74 

453 

4.14     °°S 

2,47 

115,00     ' 

57 

2.50 

2.8{.         7.44     '•" 

1  73 

454 

4.10     °°-^ 

2.48 

140.03     1 

57 

2.49 

2.89     ..((.00         " 

1.71 

0.18 

O.Ol 

57 

1. 21 

455 

9.54+°'" 

4.09 

2.49 

148.20 

2.47 

I?;!!;  o:^J 

^^     oS 

0. 70 

19.02  ,      , 

2.89 

189.87 

1.70 

45(i 

4.09     °°, 

2.50 

14'.».77+; 

57 

2.40 

2.80 

101.07+'" 

1.09 

457 

!»"      tin 

4.11+°°- 

2.51 

1.M.34     ' 

57 

2.45 

2.80 

192.27     If 

1.07 

458 

«3^     00 

<••*  ooi 

2.52 

152.91     ; 

57 

2.44 

2.90 

.93.40     \W 

194.04  ;:;■; 

105.81 

1.06 

459 

10.11     °^ 
0. 19 

10.30  , 

<18     toi 
0.07 

4.25  ,        0 

2.53 

154.48     J 

57 
57 

2.43 

2.90 

1.04 

4  no 

2.54 

150.05  , 

2  42 

2.00 

1.03 

4fil 

10.50+°-^° 

4.33  +  °°« 

2,.'->5 

157.02  +  1 

57 
57 

2.41 

19.73+°- 

2.!i0 

100.08  +  ''J 
108.14      \\'l 

1.01 

41'.  2 

10.70     °'  ° 

<•*=«     0! 

2,50 

159.19 

2,10 

20,45     °'^ 

2.90 

1.00 

403 

"'•'•'0     o'.o 

^•^'•■^     0 

2,57 

100.70 

57 

2,311 

21.19     °-lt 

2.!I0 

100.20           -^ 

1,58 

4G4 

'I'"     0:;: 

*•"«  ::;j 

4.83  ,        , 

2,58 

102.33     , 

57 
56 

2.38 

^'••'^     o.?-, 

2.90 

200.43     \\\ 

1.57 

405 

11.31 

2.5!) 

103.89  , 

2.30 

22,71  .      . 

2.91 

201.50 

1.55 

40(1 

II    r,o+0-2' 

4  q,,+o.ir, 

2.59 

105.45+' 

^6 
56 

2,35 

23,4S+°" 

2.01 

202.08-t-''' 

1,53 

407 

11.73     °-' 

b.V-J     °'» 

2.00 

ir,7  01      ' 

2,34 

24,27     °i'^ 
25,07     °-^° 
25.88     °-^' 

2.91 

203,70     '  " 

1,52 

4  OS 

11.94     °=' 

5.37     °'° 

2.01 

108.57 

2.33 

2.91 

204,00     '■" 

l.;.0 

409 

12.15     °^' 

5.59     °" 

2.02 

170.13     1 

2.32 

2.91 

200.00     '■'° 

1,49 

0.}2 

0.2J 

55 

0.82 

1.09 

470 

12.37  , 

5. 82 

2.03 

171,08 

2.31 

20,70  ^     „ 

2,01 

207.00 

1.47 

47! 

|o  r,<)+o.zi 

0.07+°'.'! 

2.04 

173,23  +  ' 

55 

2.30 

27.53 +°«-» 

28.37  °";^-» 

2,01 

208.17+'°" 

1,40 

472 

12:81     °" 

0.34     °»7 

2.05 

174.78     ' 

55 

2,29 

2,01 

200.21     ^■°l 

1.44 

473 

13,03     °-'' 

0.02     "-''^ 

2.05 

170.33     ' 

55 

2  27 

29  22     °-2 

2,00 

210..iO     '•°^ 

1.43 

474 

13.20     °-^-' 

0.91      °''' 

2.00     17 7. 87     ' 

54 

2,20 

30.0!)     °-^7 

2,90 

211.30     •■°<* 

1,41 

0.2.5 

°-3> 

I 

54 

0.88 

1.04 

475 

13.49  , 

7.22  , 

2.07 

179.41 

Ho.ns-*"' 

2.25 

31.80+°'*'' 

2,00 

212,40  . 

1  40 

470 

l.!.72+°-'-] 

7.55 +°-^.^ 

2.  OH 

54 

2.21 

2,!M)     213, 43  +  '°' 

1,30 

477 

13.95     °' 

7.90    °-." 
8.20     °-3'i 

2.0!) 

182.48     ' 

53 

2.23 

32,70     °''° 

2,00 

214.45      '•°' 

1.37 

47H 

14.18     °'^' 

2.0!)     184.01      ' 

53 

2.21 

33,07     °'^' 

2,80 

215.47     '•°' 

1.30 

479 

14.41     °'- 

8.03     °-" 

2.70     185.54     ' 

53 

2.20 

34,69     °'" 

2,80 

210.48     '■°' 

1.34 

480 

o.a4 

°J9 

I 

5» 

0-93 

0.99 

14.05 

o.oa 

2.71     187.00 

2.19 

35.52 

2.80 

217.47 

1.13 

" 

TUB 

ORBIT  OP  UUANUS. 

241) 

TABLE  IX 

Alio.  2. — Continued. 

Arjj. 

(r.y.:i) 

(r.c.3) 
3.29 

(o.i..4) 

(v.c.i) 

(p.e.O) 
407 

((..«.  1] 

(p.C.lj 

(f-x.2] 

(,.c.-> 

{,>..s.3 

(p.f.3) 

4 '20 

ft 

2.fi7 

0.40 

It 

0.42 

1957 

100 

390 

072 

128 

172 

421 

2.<!1 

;t.;!4 

0.39 

0.43 

413 

1943 

155 

399 

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129 

172 

422 

2.55 

3.39 

0.38 

0.44 

419 

1930 

150 

402 

073 

129 

171 

42.J 

2.49 

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0.30 

0.45 

425 

1910 

140 

405 

073 

130 

171 

424 

2.43 

i   3.50 

0.35 

0.40 

431 

1903 

141 

407 

673 

131 

171 

425 

2.. 37 

3.55 

0.34 

0.47 

437 

1889 

130 

410 

073 

132 

170 

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2..il 

3.(!l 

0.33  ■ 

0.49 

443 

1875 

132 

413 

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133 

170 

427 

2.25 

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0.32 

0..50 

450 

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128 

410 

673 

133 

109 

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2.20 

3.72 

0.31 

0.51 

4.50 

1848 

124 

418 

073 

134 

169 

429 

2.14 

3.78 

0.30 

0.52 

403 

1834 

120 

421 

673 

135 

108 

4:10 

2.0s 

3.S4 

0.29 

0.53 

409 

1820 

117 

424 

073 

130 

167 

4.tl 

2.o:j 

3.90 

0.2s 

0.51 

470 

1.800 

lit 

420 

673 

130 

167 

4.J2 

1.97 

3.9(i 

0.27 

0.55 

482 

1791 

111 

429 

073 

137 

lOG 

4:t;J 

1.92 

4.03 

0.20 

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489 

1777 

108 

432 

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138 

105 

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4.10 

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0.58 

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1702 

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434 

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139 

105 

435 

I. SI 

4.10 

0.24 

0.59 

503 

1748 

102 

437 

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139 

164 

4:!(5 

1.7G 

4.23 

0.23 

0.00 

510 

1733 

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440 

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140 

103 

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1.70 

4.30 

0.22 

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517 

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442 

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l.(i5 

4.37 

0.22 

0.03 

524 

1704 

90 

445 

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l.C.O 

4.45 

0.21 

0.04 

531 

1 0!IO 

94 

447 

071 

142 

161 

4»0 

1..55 

4.52 

0.20 

0.00 

539 

1075 

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143 

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441 

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91 

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159 

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1.48 

4.07 

0.18 

0.08 

553 

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1.41 

4.75 

0.18 

0.70 

500 

1031 

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145 

158 

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4.83 

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0.71 

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l(il(i 

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4(10 

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145 

157 

445 

1..12 

4.91 

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0.73 

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403 

009 

140 

150 

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1.2s 

4.99 

0.10 

0.74 

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87 

400 

608 

140 

150 

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1.24 

5.07 

0.10 

0.75 

589 

1571 

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408 

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147 

155 

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1.20 

5.15 

0.15 

0.77 

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1557 

80 

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l.ir, 

5.21 

0.14 

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80 

473 

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147 

153 

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1.12 

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1527 

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470 

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148 

152 

451 

1.0s 

5.40 

0.13 

0. 82 

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1512 

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479 

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0.80 

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88 

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0.94 

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89 

480 

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147 

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0.S5 

0.03 

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497 

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140 

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D.H-i 

0.13 

0.10 

0.94  , 

079 

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499 

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0.23 

0.10 

0.95  ' 

087 

1379 

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502 

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151 

144 

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0.77 

0.32  i 

0.10 

0.97 

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1304 

98  1 

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151 

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0.75 

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0.10 

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1319 

100 

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152 

143 

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102  1 

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1.02  ! 

719 

1320 

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512 

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405 

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llo 

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1.0(i 

713   1275 

113 

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153 

139 

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775 

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7.42 

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1203 

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533 

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155 

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7.52 

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1.35 

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1.18 

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542 

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1.50  , 

132 

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834 

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013 

15'i 

III 

479 

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8.15 

0  12 

1 .  25 

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1105  1 

103 

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1 

012 

15ii 

130 

4St) 

O.lil 

8.20 

0.12 

1.27 

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1091 

108 

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040 

150 

130 

2   July. 

lavs. 

250 

TIIK   ORBIT   OF 

URANUS. 

A 

TAULE  IX,  Aro.  2 

. — Cunliinied. 

Arff. 

'  (u.c.O)    IUff. 

yv.g.l)  l»ilT.  Sec.vnr 

iu.e.l)  Diir.  S 

cf.vur. 

(t'.».2)  Diff.  Scf.vur. 

(('.f.2)  Diff.  Si'c.vnr 

»/                     // 

tf          II 

ft 

II          II 

'» 

// 

II 

// 

II          If 

II 

4«0 

14.f.5  , 

9.02  , 

2.71 

187.00 

2.19 

35.52  , 

2.89 

217.47  , 

1.33 

4K| 

14.H9+°-^-» 

9.4:1 +°-» 

2.72 

188.57  '*"''5' 

2.18 

30.4(1  + 

3-94 

2  89 

2l8.40+°-W 

1.32 

4H3 

15.13     11^, 

9.S5     °-'' 

2.72 

190.08      ''5' 

2.10 

37.41 

3.y6 
3.y.8 

2.89 

219.44     °'^? 
220.40     °'-'l' 

l.:io 

4.S3 

'''-*T       n'^ 

'0.29     °-»^ 

2.73     191.59     '-S' 

2.15 

38.37 

2.88 

1.29 

4M 

15.r,i    °-^-t 

10.74     °-»5 

2.74 

193.09         50 

2.14 

39.35 

2.88 

221. :i5     °'^5 

1.27 

o.-M 

0.47 

1.50 

-  y.S 

0.94 

4Hr. 

15.S5  , 

11.21  , 

2.74 

194.59  , 

2.12 

40.33  , 

2. 88 

000  01) 

1.20 

4MI! 

i(;.o',» +"•-'•♦ 

ii..;9+°f 

2.75 

19.1.08  +  '-''' 

2.11 

41.32  + 

3.99 

2.88 

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1.25 

4«7 

ir,..i3    °;-» 

12.19     °5° 

2.7.1 

197..57      '•■♦^ 

2.1.1 

42.32 

1. 00 

2.88 

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1.2:1 

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l(i.5S    °-*3 

12.71     °=' 

2.77 

I99.(»5      '■•♦•^ 

2.09 

43,:!3 

I.OI 

2.87 

225;04     °'-" 

1,22 

489 

l.i..3     °]l 

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2.77 

200.53  ;;i; 

2.07 

44.35 

1.02 
1.0  J 

2.87 

225.94     y° 
0.09 

1.20 

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17.08 

13.79  ,        ^ 

2.7s 

2.12,00 

2.00 

45.38 

2.87 

220.8.3  ,      „„ 

1.19 

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17.;i3+o-^i 

14. 35+°- 56 

2.79 

203.4.1 +  '-«^^ 

2.05 

10.42  + 

1.04 

2.87 

227.71 +°'!'^ 

lis 

4:'2 

1-.5S     0-25 

n.93    "s** 

2.79 

204.92      "■-♦'' 

2.03 

47.40 

1.04 

2.8r, 

228.58     °'^7 

1.1(1 

4!».t 

17.H3     °-5 

15.52    °;'^ 

2  80 

20(1.37      '•+'' 

2.02 

48.51 

1.05 

2.8(1 

229.43     °'^5 

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0.62 

2.80 

•J07.81      '••♦•♦ 
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2.00 

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1.07 

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230.27     °Z^ 
o.«3 

1.13 

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lfi.75  ,      ^ 

2.81 

209,24 

1.99 

50.f,5 

2.85 

231.10,      „ 

1.12 

4'.M! 

ls.t;o+°-^'' 

I7.39+°-^'* 

2.81 

210,07  + '•■♦■5 

1.98 

51.73' 

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2.85 

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4'.t7 

l.s.«5     °--^ 

1,S.04     °^^ 

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,09 

2. 85 

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lit.  11     °-^ 

1S.71     °''l 

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213.50      '■■♦' 

1,95 

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0.26 

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0.70 

2.83 

214.90     '•■*° 
1.40 

1.93 

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.  1 1 
.  1 1 

2.84 

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0.77 

1.00 

5(10 

in.(i2 

20.09 

2.83     2If..30 

1.92 

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2.84 

2.35,07  ,        , 

1.04 

501 

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2.S3     217..;9  +  '-''2 

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.12 

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2.15.83  '--•■^'^ 

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21.53     °"' 

2.84     21 9.. 17     '■■' 

1.89 

58.38 

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2.83 

230.58     "■'■■^ 

1  01 

f.o:t 

20.40     °  -'■' 

00  on    °-  "4 

2.84     220.44      '•'' 

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59.52 

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2.82 

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20.  CO     °'^(' 

iiiioi    °'; 

2.84     221,81      '•■'/ 

1.80 

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2.82 

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0.9;; 

0.26 

0.76 

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0.71 

r.or. 

20.  (12 

23.77  ,       u 

2.85     223.17, 

1.85 

01.82 

2.81 

238.74  , 

0.97 

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2t.55+°J''^ 

2.85     224.52  +  '''5 

1    84 

02.98  + 

r.  16 

2.80 

2:19.44  ^°?° 

0,9.1 

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1.82 

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2.80 

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^(tS 

2  I.e.!)    °'^-' 

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2.85     227.18     '■■'•' 

1.81 

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2.79 

0,94 

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21,  It.)     °-^' 

2.1.97     °'' 

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1.79 

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2.79 

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>)■)  ■)) 

30.40      \,„ 

2.80     229.81 

1.78 

07.08 

2.7s 

242.10,      , 

0.91 

r.i2 
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2147 +°-''' 
22.72     °%^ 

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2.8(1   |23l.ll+'-'° 
2.87   i232.:l9      '-'^ 
2.87    2:i3..;7     '■" 

1,77 
1,75 
1,74 

fis  87  + 

70,(i7 

71,28 

r.19 
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.21 

2.77 
2,77 
2,70 

242.73+°- ;•' 
243.30     °   '■ 
213.98     °^' 

0.90 
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0.87 

514 

2^:21  °-''' 

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2.87   :  234.94      '  •;/ 

1.72 

72  49 

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2,75 

241.58  °-'':^ 

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,  22 

o.5,^ 

515 

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32.  H  , 

2.8S  !2:iC..20           , 

!  71 

73  71  , 

271 

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0.84 

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2:1, 75+°'^^ 

■•'•""•'"^00 

2.88     2:!7.45"'"   ■;^ 

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71,9:!  + 

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2,71 

21.5,71 -^°- 

0.83 

517 

24  00     °-j 

34.01      °'>' 

2.88     2;t8.(19         "■* 

1,(18 

7'1,  K! 

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2,73    940,20     °-^-1 

0.82 

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2».2r,    °-° 

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2.88     239. '12      '■"'■' 

1,00 

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2,72    2i'0..8O        ''■♦ 

0.8] 

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2.89  1241.13      '-■' 

1.05 

78,(1:1 

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2,72 

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0.79 

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5-.>0 

24  715 

3r..sfi  , 

2,89     212,33  ,       . 

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2.89   1  214. 7.1      ''   ■ 

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25  27     °-'-^ 

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1.00 

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2.09 

24  8,83     °-''' 

0.75 

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521 

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1 .00 

40.83      ,.„, 

2.89 ;  2(5.87   ; ;;, 

2.89     247,03            . 
'■  '.1 

1,59 
1,57 

8:1,(12 
84.88 

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2.08 
2.07 

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0,71 
0.73 

525 

2fi.0-i 

41.85  . 

2.89     248  18 

1.50 

80. 1 1 

87.41  + 

2,00 

250,21  , 

0.71 

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211.27 +°--?> 

42.88  +  '°' 

2.89     219.31 +"'3 

1  55 

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250,C,4+°  •»3 

0,70 

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2,89   [250,43      '•'■' 

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88.  CK 

2,05 

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2i;.7ii    °--* 

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2.89   1251.53      '   '■' 

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40.04      '°7 

2.89     252.. 13      '*° 

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27.72     °''-« 

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29 

2,00 

252  95     °''''' 

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50.42     ' 

2.89     25.1.90      '"S 

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2,59 

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2.89   ,257.93      '  "■' 

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2.88   ,258,95 

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54.95     '  '5 
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57.27      "^' 

2.88   ;  2.10.95     °''1 

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1.11.(13 

3" 

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2.88   '2.11.93     °'^ 

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r,;i9 

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2.88     2.12.90     °''''' 

l.:!(l 

101.25 

3' 

2.5  J 

255.02     °^5 

0,:-.5 

0.22 

1 . 1 .8 

o<)5 

'3' 

0,24 

540 

•»;..55 

58.  »5 

2,88     203,85 

1.34 

105.5(5 

2. 53 

2.55.20 

0,54 

THE   ORBIT  OF   URANUS. 


251 


* 


TABLE  IX, 

Am.  S 

.  —  Continued. 

Arj? 

(r.«.3) 

(u.f.3) 

(W.8.4) 

(0.C.4) 

1 

(p.  8.1) 

(p.C.1) 

(p.».2)  (p.c.2) 

1 

(p.«.3) 

(P.O.  3) 

480 

O.C.l 

8.2fi 

ft 
O.ll 

ff 

1.27 

851 

1001 

108 

550 

640 

156 

130 

481 

0.02 

8, 31) 

0.12 

1.28 

8(10 

1070 

173 

559 

C39 

156 

129 

482 

0.(!4 

8.47 

0.13 

1.30 

808 

1003 

179 

501 

037 

156 

128 

483 

o.i;5 

8.58 

0.13 

1.31 

877 

1040 

184 

504 

030 

156 

128 

484 

0.«t! 

8.  GO 

0.14 

1.33 

885 

1035 

190 

500 

634 

156 

127 

4s:> 

0.(!7 

8.79 

0.14 

1.35 

894 

1021 

190 

509 

033 

156 

120 

4Hr. 

0.1.0 

8.00 

0.15 

1.3fi 

902 

1007 

202 

571 

632 

156 

125 

4S7 

0.71 

O.Ol 

0.15 

1.37 

911 

094 

208 

574 

630 

156 

124 

4t»S 

073 

0.11 

0.1(1 

1.39 

919 

980 

214 

570 

629 

166 

124 

480 

0.75 

9.22 

0.17 

1.40 

927 

900 

220 

579 

627 

156 

123 

400 

0.77 

0.33 

0.17 

1.42 

930 

953 

227 

581 

620 

156 

122 

401 

0..80 

0.43 

0.18 

'..43 

945 

040 

234 

584 

024 

156 

121 

402 

0.82 

0.54 

0.10 

1.44 

954 

920 

241 

580 

023 

150 

120 

'0:t 

0.85 

0.(15 

0.10 

1.4(1 

903 

913 

249 

580 

021 

150 

120 

;i>4 

0  88 

9.75 

0.20 

1.47 

972 

000 

250 

501 

619 

150 

119 

4  or. 

0.01 

9.8fi 

0.21 

1.40 

081 

887 

204 

504 

018 

150 

118 

40(; 

0.04 

9.07 

0.22 

1 .  50 

000 

874 

272 

500 

0 1  (1 

150 

n7 

407 

0.08 

10,07 

0.23 

1.51 

090 

801 

280 

509 

015 

150 

117 

4  OS 

1.02 

10.18 

0.23 

1.53 

1008 

848 

2HH 

001 

013 

155 

no 

400 

lot! 

10.29 

0.24 

1.54 

1017 

835 

290 

0O4 

on 

155 

115 

500 

1.10 

10.39 

0.25 

1.55 

1020 

822 

304 

000 

010 

155 

115 

501 

1.14 

10.50 

0.2(1 

1.57 

1035 

810 

312 

008 

008 

1.55 

114 

502 

1.10 

10.  CO 

0.27 

1.58 

1(144 

707 

321 

on 

000 

155 

114 

503 

1.24 

10.70 

0.28 

1..59 

105:) 

785 

320 

013 

(104 

154 

113 

504 

1.20 

10,81 

0.29 

1.(10 

1003 

772 

338 

615 

002 

154 

113 

50;-. 

1.34 

10,01 

0,30 

l.r>2 

1072 

700 

340 

f.l7 

000 

154 

112 

501', 

1.30 

11, (tl 

0.31 

1.(13 

1081 

748 

355 

020 

598 

154 

112 

607 

1.44 

11.12 

0.32 

1.(14 

1 000 

7.30 

304 

022 

500 

154 

111 

50S 

1.50 

11.22 

0.33 

1.(15 

1009 

724 

37.{ 

021 

504 

154 

11(1 

500 

l.afi 

11.32 

0.34 

1.(1(1 

1108 

712 

382 

020 

502 

153 

llo 

510 

1.(12 

11.42 

0.35 

1.07 

1117 

700 

391 

028 

500 

153 

109 

511 

l.<;8 

11.52 

0.37 

i.r.8 

1120 

088 

401 

031 

588 

153 

109 

512 

1.74 

ll.ti2 

0.38 

1.(19 

1135 

077 

410 

033 

585 

153 

108 

51.t 

1.80 

11.71 

0.30 

1,70 

1114 

000 

420 

035 

583 

153 

108 

514 

1.87 

11.81 

0.40 

1.71 

11.53 

054 

430 

037 

581 

153 

107 

515 

1.04 

11  01 

0.41 

1,72 

Il(!2 

043 

440 

030 

578 

1,53 

100 

5111 

2.01 

12,00 

0.42 

1,73 

1171 

032 

450 

042 

570 

153 

100 

517 

2.08 

12.00 

0,43 

1  74 

IHO 

021 

40(1 

014 

573 

152 

100 

518 

2.  Hi 

12.10 

0.45 

1,75 

1189 

010 

471 

040 

571 

152 

105 

619 

2.23 

12.28 

0.4« 

I.7fi 

1197 

599 

481 

048 

500 

152 

105 

520 

2.31 

12.37 

0.47 

1.77 

1 20(1 

588 

403 

050 

.500 

152 

104 

■,il 

2. 30 

12.111 

0.40 

1.78 

1215 

577 

503 

052 

504 

152 

104 

522 

'i■^^ 

12.5.-. 

0.50 

1.70 

1221 

507 

514 

051 

501 

151 

103 

52;j 

2.55 

12.(13 

0.51 

1.70 

1 2:!3 

557 

525 

050 

558 

151 

103 

524 

2.(!3 

12.72 

0.52 

1.80 

1242 

540 

530 

058 

550 

151 

103 

525 

271 

12.81 

0.5 « 

1,81 

1250 

530 

548 

000 

553 

151 

102 

52(i 

2.80 

12,80 

0.55 

1,82 

1  250 

r,..,! 

550 

002 

550 

151 

102 

527 

2.88 

12.08 

0.57 

1,83 

12117 

..17 

571 

004 

518 

150 

101 

528 

2.07 

1  1.0(1 

0.58 

1,83 

1 270 

5(17 

583 

000 

515 

150 

lot 

520 

3.0ti 

l.t.14 

0.(10 

1.84 

1281 

408 

595 

008 

5(2 

ho 

100 

5110 

3.15 

13.22 

O.CI 

1.85 

1203 

488 

fi07 

07O 

540 

15(1 

100 

5,{1 

3.24 

13  :10 

0.(12 

1  85 

1   . 

479 

018 

(;-2 

537 

150 

100 

5,12 

3.;' 4 

13.37 

n,(it 

1,8(1 

1310 

470 

O.'iO 

073 

535 

U'.l 

00 

5;i:i 

3.43 

13.45 

0.(15 

1.8(1 

1318 

4111 

042 

075 

532 

140 

00 

634 

3.53 

13.52 

0.(17 

1.87 

1327 

452 

064 

077 

520 

149 

08 

535 

3.03 

13.59 

0.fi8 

1.87 

1335 

443 

000 

078 

527 

140 

98 

53r. 

3.73 

13.(1(1 

0.(10 

1.88 

1313 

43  4 

078 

080 

521 

148 

98 

537 

3.83 

13,73 

0.71 

1.88 

1352 

420 

(100 

0S-.< 

521 

14s 

08 

538 

3.03 

13,8(1 

0.72 

1.80 

I.'IOO 

417 

7(12 

083 

5 1 !» 

lis 

07 

530 

4.03 

13,87 

0.71 

I.SO 

13HS 

400 

711 

osr. 

510 

I4S 

07 

510 

4.13 

13.o;i 

0.75 

1.00 

1370 

401 

72'1 

osci 

513 

117 

07 

35:; 


THE   OUBIT  OF   URANUS. 


11 


TAHI.K  IX,  Ami   i 

.  —  (Unwluili'd. 

Arj5. 

(D.c.O)    Diff. 

(u.D.l)  Diir.  Sfc.var 

(c.p.l)  Diir.  !• 

M'c.viir. 

I  i'.(i.2)  Diff,  Sec.vnr 

'(i^.c.2)  Diff.  Sic.vnr. 

540 

29.55 

58.45  ,  ,     „ 

2.88 

203.85 

1.34 

105.50 

If 

2.53 

2.55.20  , 

0.54 

541 

L>9.77+°" 

59.  ('.3 +  '''^ 

2.S8 

204.79 +°-9^ 

1.33 

100.88  + 

•3» 

2.52 

2.-.5.49+°-' 

0.53 

543 

29,99     °'^' 

00.  X2     '•"' 

2..S8 

205.71     °'^' 

1.32 

108.19      ' 

■,i' 

2.r,l 

:2.-.5.70    °-' 

0,52 

543 

30.-.>l      in 

02.02      ■•-° 

2.87 

200.02     °-'^' 

1.30 

109.51      ; 

•,»J 

2.50 

2.M.90    °;° 

,250.08     °''^ 
0.17 

0,51 

544 

30.43     °- 

03.24    ;•;- 

2.87 

207.52     l;^ 

1.29 

110.83      J 

■32 
•3^ 

2.49 

0.50 

545 
54fi 
547 

••M.O.;    ll\ 

04.40  , 
05.09  +  ' -'3 

00.93      ••-'♦ 

2.87 
2.87 
2.87 

208.40  ,      „ 

.,,;,,  07+O.S7 

270.' ?2     °-':;5 

1.27 
1.20 
1.25 

112.15  ,, 
113.47+1 
114.80     , 

3-' 
•33 

2.47 
2.40 
2.45 

250.25  ,        , 
2.Ml,41+°"^ 
250..55     °'-* 

0.49 
0.48 
0.47 

548 
549 

08.18  ;•-' 

09.43      \]l 

2.80 
2.80 

270.90     °-'^-* 
271.79     °'^3 

1.21 
1.2  J 

110.12 
117,45     J 

■3-s 

32 

2.44 
2.43 

2,-.0.08     °'3 

250.79     I'' 
0. 10 

0.40 
0.45 

550 
551 

0. 19 

71.90+   -' 

73.23  \:i 
'-«o  ,:3o 

2.80 
2.80 

272.00  ,  „  „ 
273.40+°-^° 
•--T4.I8     °! 
271.95     °t^ 

1.21 
1.20 

118,77  ,, 
120.09'^, 

32 

2.42 
2.41 

2.J0.S9  , 
2.V198+°°9 

0.03 

0.44 
0.43 

652 
5.".3 
554 

2.85 
2.85 
2.84 

1,18 
1,17 
1.15 

121.42     , 
122.74 
124.07     , 

32 
32 

2,40 
2.39 
2.38 

0.42 
0.41 
0.40 

555 

32.f.3  ,        - 
32.Sl+°''^ 

33:17  °;« 

77.10  , 

2.S4 

270.41 

1.14 

125,39  , 

2.  .30 

257,18  , 

0.40 

550 

7«.40  +  '--'° 

2.84 

277.10+°-7^ 

1  12 

120.71+' 

32 

2.35 

257,20+°°* 

0.39 

55T 

79.71     •••'■ 

2.83 

277.80     °^^° 

1.11 

128.03     ' 

3-' 

2,31 

2.-.7.2>     °°° 

0,38 

558 

81.03      '••5^ 

2.83     27  s.. 55     °'"} 

1.09 

129.35     ' 

3-' 

2.33 

257, 1 9 -'^■°' 

0,37 

559 

33.35  °'^ 

82.35      '-3- 

2.82 

279.23     °"'; 

1.08 

130.07     ' 

3-' 

2.32 

257.10     °°3 

0.30 

0.17 

>•.?,? 

0.66 

34 

0.04 

5f;o 

33.r.-.>  , 

83.08  , 

2.82 

279. «9 

1. 00 

132.01 

2.31 

257,12 

0.35 

5i;i 

33.(;9+°-'7 

85.01+'-" 

2.82 

2s().54+°-''5 

1.(15 

1:13.33  +  ' 

32 

2.30 

2.--7,o7-°-°-; 

0.34 

5(!2 

33.Hfi     °-'7 

80.-35      '•■'•* 

2.81 

2si.n     °''' 

1.03 

131.05 

32 

2  29 

257,00     °°l 

0.33 

6(;3 

34.93     °'; 

34.19     °'^' 
0.  :6 

87.09     '■•'■* 

2.81 

2S1.78     °-^' 

1.02 

135.97 

3- 

2.27 

0.33 

5U4 

89.04    '-3; 

i,,;6 

2.80 

2S2.38     °-^° 
°-59 

1.01 

137.28      \ 

3* 
32 

2.20 

2.-,o',s2     °  '° 
0, 1 1 

0.32 

5r.5 

34  .n5  ,        , 
31.51 +°'f' 

90.40  ,         , 
91.70+'-"^ 

2.80 

282.97 

0.99 

13^,00 

2.25 

-•^'■•''   0,, 

0.31 

5i;(i 

2.79 

2s3.54+°-5' 

0,98 

139.91 +, 

.1  ( 

2  21 

2.-,0  59-°" 

0  30 

5<i7 

9.1.13    ;••" 

2.79 

?   4.09     °" 

0,97 

141.2? 

3( 

2,23 

2.-,o45  °;^ 

0.29 

5(18 

34. S2     °''' 

91.50         -7 

2.78 

2^4.03     °-'^-* 

0,90 

142,54 

32 

2.21 

250,  .30     °'5 

250.14     °;^ 
0.  la 

0.29 

5(19 

34.97     °'5 
o.is 

2.78 

2M5.10     °-^-^ 
°-5' 

0.94 

143,s5     J 

3' 
3' 

2.20 

0.28 

5T0 

35.12  , 

3.'..27"^°-'5 

9><.04+     3 

2.77 

2S5.07  , 

9.93 

145.10  , 

2.19 

255,90 

0.27 

!)71 

2.70 

2S0.10+°I2 

■1.92 

140.40  +  ! 

30 

2.18 

255,77-°:^ 

0.20 

572 

3.V41     l[-\ 

1(111.03      '••''^ 

2.70 

280.04     ^-l:' 

0.90 

147.70     J 

30 

2.10 

2.V..57     °-   ° 

0.20 

573 

3:V5:,    °-  -^ 

101.42      '••" 

2.75 

2S7.10     °-''^' 

0.89 

14!1.05 

29 

2.15 

255.30     °- 

0.25 

574 

O.I.^ 

I()2.S2      '■•'° 
1.40 

2.75 

2S7.55     "•■'■^ 
0 -I,? 

0.87 

1^.0,35     , 

30 
29 

2.13 

^^^•'^  :.]i 

0.25 

575 
57(1 

35.95+°  '• 

101.22  , 
l(l.-,.r,2  +  ''° 

2.74 
2.73 

287.98  , 
28^.40+°'- 

0.8fi 
0.85 

151.01  ,, 
l-.2.9::"^' 

20 

2.12 
2.1] 

25 1 , S9           r 
254,03-°:; 

0.24 
0.23 

577 

3n.08     °'-^ 

107.03    '•;'' 

2.73 

2'.!^  so     °-^° 
2S9.18     °-'''^ 

0.S3 

154,22 

2') 

2.09 

2.-i4.30     ^-^l 

0.23 

578 

3(;.2i    I'l 

10S.44     [^' 

2.72 

0.W2 

155.50 

2.08 

0. 22 

579 

3(i.33 

0.12 

109.85     J-^J 

2.72 

289.55     °-'J 
o.j6 

0,80 

150,77      , 

-'7 
28 

2.00 

^^^'^"  :.^ 

0.22 

5>*0 
5S1 

3(;.(;9   °'-' 

3.;.s,  °;-: 

'"■'-"4-1   12 
112.08+    ■■' 

lU.lO     ["[l 

11.-..52     •;^ 
110.95    ;;]j 

2.71 
2.70 

289.91   , 
290.25 +°-'^ 

0.79 
0,78 

158.05  ,, 
159.32''" 

27 

2.05 
2.01 

-•'"■••«_o  V 

0.21 
0  21 

5^2 
5^3 

2.09 

2.08 

290.57     °-;' 
290.  S8     °''' 

0,77 
0,70 

100. .-o 
101,S5 

27 

2,02 
2.01 

0.20 
0.20 

684 

3.1.93  °;; 

2.07 

291.18  :-^ 

0,75 

103,11      , 

26 

1.99 

0.19 

585 

37.04 

118.38  , 

2,07 

291.40 

0,73 

104,37 

1.98 

251,75 

0.19 

58(5 

37.1:. +°" 

119.81 +'•'»•' 

2.00 

291.72+°-'^ 

0  72 

105,02  +  ' 

2,i 

1.97 

251, 30 -°-''' 

0.18 

5.S7 

37  2fi     °- ' ' 

121.24     '•■'■? 

2.05 

291.90     °-* 

0,71 

100,v7      ' 

25 

1.95 

250,90     °-'° 

0.18 

5S8 

37.;;7   °" 

122.07      '    '■' 

2.04 

292.19     °  -■' 

0,70 

KWII      ' 

■4 

1.94 

2.-.0.55     °-t' 

0.17 

589 

87.48     °" 

0. 10 

121.11      '^-^ 
>-43 

2.03 

292.41     °-' 

0.  20 

0.09 

109,35      ' 

-M 

1.92 

2.-10,13     °-" 
0.44 

0.17 

5'10 

37.59  , 

125.54  , 

2.02 

292.01 

O.OS 

170.58 

1.91 

219.09 

0.10 

591 

37.(;>s+°'° 

120. 98  +  '-'-* 

2.(11 

292.79 +°''^ 

0,07 

171, H1+' 

23 

1.90 

249.24-°-;^ 
248,78     °-t'^ 

0.10 

592 

37.78     ^'^ 

128.41      '•« 

2.00 

292.90     °  'J 
293.12     °' 

0,00 

173,03      ' 

,  22 

1.88 

0.15 

593 

37.88     °' 

129.85      '  •♦^ 

2.59 

0.04 

171,25      ' 

,22 

1.87 

248,31      °-*l 
217.83     °-»** 

0.15 

594 

37.98     °'° 

131.29      '••♦■♦ 

2.58 

293.20     °''* 

0.03 

175.40      ' 

.21 

1.85 

0.14 

0. 10 

1.44 

0.12 

.21 

o.";o 

595 

3S.08  , 

132.73,      ,, 

2.58 

293.38 

0,02 

170.07  , 

1.84 

217,33 

0.14 

59A 

3S.18+°''" 

,,.,.,7  +  1.44 

2.57 

293.49+°" 

0,01 

177. X7+' 

.20 

l.«3     240  S2- 5 

0.14 

697 

3><  27     °  °'' 

135.01      '•••♦ 

2.50  I293.5X     °°'> 

0.00 

17".t.(t7 

.  ,;o 

l.Sl     210,30     °-''' 

0.13 

598 

38.30     °"'' 

1. •17.05      '    '* 

2.55     293  00     °°'^ 

0,5  s 

1HO,20      ' 

II) 

1,S0     21577     °  •'^•' 

0.13 

699 

3S.45    "■'"'  i;is..-,o    '  •'^ 

2  51     293.73     °"' 

0.57 

1><I,U      ' 

,  iS 

1,7s     21,-,. 2:1     °%» 

0.12 

o.fv                      i.K 

1                  0  oi 

.  iS 

(too 

i    38.54             ,l:!9.n4 

2.53   ,2;i3.78 

0.50 

1H2,0J 

1,77     214,07 

0.12 

: 


THE 

ORBIT  OF 

URANUS 

• 

253 

TAB 

.K  IX,  Alio.  2.—ComUided. 

Arg. 
.')40 

(y.«.3) 

{v.c.H) 

(0.8.4) 
ft 

0.75 

i(u.c.4 

(^.r.O 

(p.-S.lJ 

(p.f.lj 
720 

(p.<..2 

(p.c.2 

(,).«.3, 
147 

[  (p.e.3 
97 

4.13 

13.93 

1.90 

1370 

401 

080 

513 

f>4l 

4.23 

13.99 

0.70 

!  1.90 

I3H1 

393 

738 

088 

510 

147 

97 

542 

4.34 

14.05 

0.78 

1.90 

1392 

380 

751 

089 

507 

147 

90 

54:{ 

4.45 

14.11 

0.79 

1 .  90 

1400 

a78 

703 

091 

503 

140 

90 

544 

4.50 

14.17 

0.81 

1.91 

1408 

371 

770 

092 

500 

140 

95 

545 

4.07 

14.23 

0.83 

1.91 

1410 

303 

7S8 

093 

497 

140 

95 

54ti 

4.79 

14.28 

0.84 

1.91 

1124 

350 

801 

095 

491 

140 

95 

547 

4.90 

14.34 

0.85 

1  1.91 

1431 

319 

813 

090 

491 

140 

94 

54S 

5.01 

14.39 

0.87 

1.92 

1439 

343 

820 

098 

4S7 

145 

94 

549 

5.13 

14.44 

0.89 

1.92 

144(; 

330 

t'38 

099 

484 

145 

94 

550 

5.24 

14.49 

0.90 

1.92 

1454 

329 

851 

700 

4SI 

145 

93 

551 

5.35 

14.54 

0.92 

1.92 

1401 

323 

804 

702 

47s 

145 

93 

552 

5.47 

14.58 

0.93 

1.92 

1408 

310 

877 

703 

475 

145 

93 

55;{ 

1   5.58 

14.03 

0.95 

1.92 

1475 

310 

890 

704 

471 

144 

93 

554 

5.70 

14.07 

0.90 

1.92 

14S2 

304 

903 

705 

408 

144 

92 

555 

5.82 

14.71 

0.98 

1.92 

14S9 

298    910 

7ni; 

405 

144 

92 

55r> 

5.94 

11.75 

0.99 

1.92 

1  490 

292    929 

70s 

402 

I4t 

92 

55  T 

0.00 

14.79 

1.01 

1.92 

1502 

2H7    913 

709 

45S 

141 

92 

558 

0.18 

14.82 

1.02 

1.92 

1509 

2S1    950 

710 

455 

143 

92 

559 

0.30 

14.85 

1.04 

1.92 

1515 

270 

970 

711 

452 

143 

91 

500 

(5.42 

14.88 

1.05 

1.92 

1522 

270 

983 

712 

449 

143 

91 

5r,l 

0.54 

H.91 

1.07 

1.92 

1528 

205  w.h; 

71.i 

445 

143 

91 

502 

0.07 

11.93 

1  08 

1.92 

15.J4 

200 

1010 

711 

4  42 

143 

91 

5G:J 

C.79 

It. 90 

1.10 

1.91 

1541 

255 

1(23 

715 

438 

142 

91 

5G4 

6.91 

It. 99 

1.11 

1.91 

1547 

250 

M:!i; 

710 

435 

142 

90 

505 

7.04 

15. ni 

1.13 

1.91 

1.553 

240   10(1) 

710 

431 

142 

90 

500 

7.10 

15.(i:l 

1.14 

1  91 

15.-.9 

242   lOi;:! 

717 

428 

142 

90 

507 

7.29 

15.05 

1.10 

1.90 

15r,4 

2.3  s 

lOTO 

717 

421 

142 

90 

508 

7.41 

I5.0fi 

1.17 

1.90 

1570 

231 

lOSO 

717 

421 

HI 

90 

509 

7.54 

15.08 

1.18 

1.90 

1575 

230 

1102 

718 

417 

141 

89 

570 

7.00 

15.09 

1.20 

1.S9 

1581 

220 

1110 

7is 

413 

141 

89 

571 

7.79 

15.09 

1.21 

1.89 

I58(i 

22.3 

112:1 

718 

410 

141 

89 

572 

7.91 

15.10 

1 .  23 

1.89 

I5:il 

219   1112 

71S 

400 

Itl 

89 

573 

8,04 

15.10 

1.24 

1.88 

I59i; 

210   1155 

719 

4o:t 

140 

89 

574 

8.10 

15.11 

1.20 

1. 88 

1001 

213 

1109 

719 

399 

140 

89 

5T5 

8.29 

15.11 

1.27 

1.S7 

1005 

210 

1182 

719 

395 

140 

88 

570 

8.42 

15.12 

1.29 

l.Sfi 

1010 

207 

1195 

719 

392 

no 

88 

577 

8.54 

15.12 

1  30 

1.80 

i<;i4 

205 

1208 

719 

388 

HI) 

88 

57  S 

8.07 

15.11 

1.32 

1.8.T  ' 

1018 

202 

1221 

719 

3S.^, 

140 

88 

679 

8.80 

15.10 

1.33 

1.84 

1022 

200 

123  4 

719 

381 

139 

88 

5  SO 

8.93 

15.09 

1.34 

1  84 

1020 

19S 

1218 

719 

37  s 

139 

88 

5SI 

9.05 

15.08  j 

1.35 

I. S3  j 

1 030 

19i; 

1 21;  1 

719 

371 

139 

87 

5S2 

9.18 

15.07 

1.30 

1.8.1 

10:!4 

19 1 

1271 

719 

371 

l;!9 

87 

5s;} 

9.31 

15.05 

1.38 

1.82 

ir,:{7 

I9i 

JOsIS 

719 

307 

139 

87 

584  i 

9.43 

15.04 

1.39 

1.81 

1041 

191 

1301 

719 

304 

139 

87 

585 

9.50 

15.02 

1.40 

1.91 

1045 

ISO 

1315 

718 

300 

139 

87 

680 

9.08 

15.00 

1.42 

1.80 

HUS 

188  : 

1328 

718 

357 

139 

87 

587 

9.81 

14.98 

1.43 

1.79 

1051 

IM.; 

1342 

718 

35:1 

139 

87 

588 

9.93 

14.90 

1.44 

1.79 

1054 

185 

1350 

717 

350 

138 

80 

589 

10.00 

14.93 

1.45 

1.73 

1057 

184 

1309 

717 

340 

138 

80 

590 

10.18 

14.91 

1.40 

1.77 

1000 

18.3 

1382 

717 

343 

138 

sr, 

591 

10.31 

14.88 

1.48 

1.70 

|i;(;2 

1S2 

1390  j 

710 

339 

138 

80 

592 

10.43 

1  t.s5 

1.49 

1.75 

1005 

1S2 

1409  1 

710 

335 

138 

80 

593 

10.55 

14.81    : 

1.50 

1  74 

1007 

isi  , 

142i 

710 

332 

138 

80 

594 

10.08 

14.78  ; 

1.51 

1.73   1009  1 

181 

1430 

715 

328 

138 

85 

595 

10.80 

14.74 

1.52 

1.72 

1071 

ISl 

1448 

715 

324 

138 

85 

590 

10.92 

14.71 

1.53 

1.72   1073 

181 

HOI 

711 

321 

138 

85 

597 

11.04 

14.07  i 

1.55 

1.71   1071 

181 

1t7t  i 

711 

317 

137 

85 

598 

11.10 

14.03 

1.50 

1.70   ICTO 

ls| 

14S7 

713 

313 

137 

85 

599 

11.28 

14.58  i 

1.57 

1.09   11,77 

182  ! 

1500 

712 

310 

137 

85 

fiOO 

11.40 

14.53 

1.58 

1.08   1078 

182 

1513 

711 

300 

137 

85 

t 


254 


TIIK   ORBIT   OF    URANUS. 


[':; 


TAlil.K  X,  Alio.  3. — Aitupn  otf  Nkiti  nk. 

Ar-. 

(f.r.O)  Diir. 

(I'.-i.l)    Diir. 

(r.c.l)    Diir. 

(r.«.2)  I)iir.(,iv.2)  Dill'. 

(r.«.3)  (I'.r.W)  (r.^.i) 

,  (•••'••4) 

„ 

#f                  tl 

/»          // 

M                  tt                     tt                 ft 

tf 

It 

tf 

tl 

0 

02.  Of. 

0.87  , 

31.22  , 

7.42           ,   3.05 

0.80 

1.15 

—0.85 

—  1.13 

1 

ii2.r.7-°"^ 

0.08 +°'" 

31.77 +°-^S 

7.30-°°^'   3.82-°-'-' 

O.sl 

I.IO 

O.SO 

1.14 

i 

02  38     °-'^ 

1.10     °'^ 

35.31     °-S-» 

7.;iO     °°'^  3.70     °'^ 

0.X3 

lis 

0.S7 

1.15 

3 

02.07     °-'' 

1.23     °'-l 

35.S5     °-5« 

7.23     °°7  3.58     °-'- 

C.S4 

1.20 

0  SO 

1.10 

4 

»'•"  I:]: 

0.15 

30.39     ::M 

7.10     °°7  3.47     °" 
0.07                O.II 

U.80 

1.22 

O.OI 

1.17 

5 

01. 4n 

1.53  , 

30.02  , 

7.00           ..   3.30 

0.S8 

1.23 

—0,03 

—  1.18 

(> 

01.15-°'' 

1.70+°-'7 

37,41 +°--^=' 

7.01 -°°''   3.L'5-°" 

U.  0(1 

1.24 

(»,  !I5 

1.18 

7 

0O.S4     °''' 

l.ss     °'"' 

37.00     °-5-' 

0.02     °°''   3.15     °'° 

0.01 

1.25 

0.07 

1.10 

s 

00.52     °  •^'' 

-'•"■^     0'.' 

3.48     °-5^ 

0.S2     °'°  3.05     °'"^ 

0.03 

1 .  20 

1.00 

1.19 

9 

00.19  :^ 

2  29     °'" 

30.00     °-"' 
0.51 

0.72     °'°   2.05     °'° 
O.II                o.oy 

0.05 

1.27 

1.02 

1.20 

10 

so.sfi 

2.51  , 

30.51 

0.01                2.  SO 

0.07 

1.28 

—1.04 

—  1  20 

11 

H0.53— °--^' 

2.75 +"•-''» 

4(l.(i2+°-5' 

0.51 -°"°  2  77-o-oy 

1.00 

1.28 

1.00 

1.19 

12 

SO.  18     °-3S 

2.  CO     °-t 

40.52     °-5° 

0.40     °"    2.00     °-°''* 

1.02 

1.2:. 

1.08 

1  10 

l.'t 

HA.H3      °'^'' 

3.20     °-7 

41.01     °-W 

,.    ojj          0.  I  J      .,    ,;|          0.08 

1.05 

1,29 

1.00 

1  18 

14 

88.47     °-''' 

3.54     °-''' 

41..,0     °-l'> 

oilO     °'-l2;54     °-°7 

1.07 

1.29 

1.11 

1.18 

0.3S 

0.J9 

o..»y 

0. 12 

0.07 

1") 

8*<.00 

3. S3  ,      „ 

41.00, 

0.04 

2.47 

1.00 

1.29 

—  1.12 

—  1.17 

ir> 

K7,7,-°-3^ 

4.13+°-*° 

4.J  .(7+0.4S 

5.01 -°'-^ 

2.41-°°" 

1.12 

1.20 

1.13 

1.15 

17 

H7.31     °-«° 

4.15     °i' 

4-1 04     °-»7 

5.78     °'3  2.;i,i     °°5 

1.14 

1.2s 

1.15 

1.13 

IS 

8r.,oo    °-*' 

4.7s     °-53 

4.!.  40     °-tf 

5.04     °'-«   2.31      °'°5 

1.17 

1.27 

1.10 

1.12 

1!) 

80.49     °»' 
°-4.5 

6.13  °y= 

0.3O 

43.80     °-*^ 
0.45 

5  50     °'t   2  "7     °°'' 
0. 141  "'"       0.04 

1.19 

1.20 

1  17 

1.10 

20 

8(1.0(5 

5.40  , 

44.31  , 

5.30             !2.23 

1  21 

1.25 

—  I. is 

—  l.OS 

k>| 

s.-,.r,2-°'',' 

5.s7+°-''^ 

4,.;, +0.4.1 

5  22—°''|2  •.'0-°°-J 

1.23 

1.23 

1.10 

1.00 

'2-2 

85.10     °-t'' 

0.2(1     °-''^ 

45.10     °-*^ 

5.08     °"   2.18     °°' 

1.25 

1.22 

1.10 

1.03 

'J.l 

84.00     °-'7 

0.(;7    °-i' 

45  58     °-i- 

4.03     °'-^    2.10-°°* 

1.27 

1.20 

1.20 

1.00 

a  4 

84.22     °-'7 
0.50 

7.00     °''-' 
0.4.) 

45.0s     °--l° 
0.40 

4.7s     °'-^    2.10     °°° 
0.15                0.00 

1.29 

1.10 

1.20 

0.00 

2r) 

83.72 

7.53  , 

40.3Sj^ 

4  03               2.10 

1.31 

1.10 

—  1.20 

—0.07 

•jc. 

8:!.22-°-5'' 

7  !is+°-»5 

4C,.-T+°-'^'' 

4.48-°- "S   2.10     °'°° 

1.33 

1.11 

l.lOi 

0.05 

27 

82.70     °-^' 

S.I5     °-i' 

4T.lt     "■•7 

4  n   °'^  '»  17+°'°' 

1.35   1 

1.12 

1.101 

0.0;! 

2S 

82.17     °-^' 

s.oi    °-'" 

47.10  °-y> 

lilS     °'-V2:iO     °°- 

1.30 

1.10 

lis 

O.itl 

2'.» 

81. 02     °-^-^ 

0-57 

0.  ^0 

''■''     0.5. 

47. S3     °-'t 
033 

j  „.)     oif'  .,  .,0     0.03 
'   '     o.iSi""       0.04 

1.37 

1.07 

I.17j 

0.00 

SO 

81.05 

0.05  , 

48.10  , 

3.S7           ■  I2.20 

1.38 

1.05 

-1.17 

—0.89 

:!l 

80.48—^''' 

,„  ,^+°o,5 

4s.i7+°-'' 

.;  r.i-o'.i    .,  .,„4-o°4 

1.3s  ' 

1,02 

1.15 

0.88 

»4 

70.  SO     °P'' 
70.20     °  '° 

11 5s  °-:'.' 
12.15  °:' 

0.5.S 

4..7,;    "•=:' 
4:1.. 11   °-'; 

40.. 10     °-''^ 
o.2j 

3:.5S    °'-t  2;35    °°s 

3.13     °'-^    2.11      °°'^' 

3.2s     °'-='    2.4S     °°7 
0.14               0.07 

1.30 
1.40 
1.40 

1,00 
0  07 
0.01 

1.13 
111 

1  00 ; 

O.SO 
0.  S5 
0.S4 

35 

7S.05        , 

12.73 ,    , 

40.55  . 

3.14               2  .55  , 

1.40 

O.OI 

—  1.00 

—  0.83 

:<)> 

(7.41                  y 

l:i.:!3+°?" 

4'..77+°-=-' 

3.01-°''  2.0l+°-°9 

1.40 

o.ss 

1.07 

0.S3 

.•a 

3'J 

■Hi  Hi 

0.65 

40.  OS     °-^' 

50.10     °';^ 

50.32     °'^ 
0.14 

0.14  „-.,     0.09 
•"74     °'-'  ■>'s:i    °'° 

201  °"' :';o4  °-" 

0.12                      O.II 

1.40 
1.39  ■ 
1.38 

0.S5 
O.S-J 
0.70 

1.05 
1.03 
0.00 

0. 82 
0.S2 
0.81 

40 
41 
4-2 
43 
44 

>^-s"  C.70 

S^s+°- 

2.10                3.05  , 
03s -°"    3  17+°" 
2.27     °"    3  30     °'' 
2  17     °'°   3  14     °'-' 

•..37 
1.35 
1.34 
1.32 
1.30 

0.70 
0.73 
0.70 
0.07 
0.05 

—0.00 
0  07 
0.05 
0.03 
0.01 

—0.81 
O.sl 
0.S2 
0.S2 
0.83 

45 

71.14 

10.20  , 

50.  S3 

1.07             !3'^,„,. 

1.28 

0.02 

—0.01 

—0.83 

4ri 

70.39 -°-75 

10.07+°- 7' 

50.  S3     °-°° 

1  ss-°°''  3.H7+°-'5 
isl     °°7   4.o:t     o-*(> 
1  -.     0.00    ,  |i)     0. 10 

,;.;;  0.06;^;.,;.  0.17 

1.25 

0.00 

0.91 

0.84 

47 

00.03     °''' 

2(1.  OS     °"' 

50.sl-°-°;' 

1.23 

0.58 

0.S7 

O.SO 

4S 

OH.  87     °'7*' 

21.40     ^^-7-' 

50.75     °°'^ 

1.20 

0.50 

0.80 

0.S7 

40 

08.10     °" 

22.12     °'- 

50.07     °°'^ 

1.17 

0.54 

0.80 

0.88 

0.78 

07.5 

o.io               0.0^                0.17 

50 

07.32 

22  S5 

50.57                1.04           ■    4. .53 

1.14 

0.52 

—0.84 

—0.00 

51 

(•,0.54 -°''^ 

23.5S+°-7' 

.50.11-°  '.'    1.01 -°°-^   4.71+^-'!^ 

1.10 

0.51 

0.84 

0  02 

52 

05.75     °" 

oj    j.j     0-74 

50.20     °'5 

1.5S     °°-^   4. so     °''^ 

1.07 

0.40 

0.83 

0.04 

53 

04.00     °-7" 

25.(10     °"> 

50.11  °''^  1.50  °°^  5.(»7  ° '■;; 

1.03 

0,4  s 

0.82 

0.00 

54 

04.  IP,     °''° 
o.Si 

25.70     °-7.' 
0.74 

40.00     °"    1..55     °°'   5.25     °''^ 
0.24                0.00                0.  iS 

1.00 

0.47 

0.81 

0.98 

55 

03.. 35 

20.53  , 

49.00                1.55  .             5.43  . 

0,00 

0.47 

—0.81 

—  1.00 

5fi 

02.54 -°-^' 

27.27+°- 7' 

40.:io-°-^7 

1 57+°°'  5.02+°"' 

0.02 

0.47 

0.81 

1.01 

5t 

01.7.3     °f 

27.00     °"- 

40  11      °-'^ 

1.50     °°-   5.SI     °") 

ix.i    °-°-'  5.00    °'« 

1.00     °°,',0.I7     °''' 

O.sO 

0  47 

0.  S2 

1.03 

5S 

00.02     °''' 

1    OS  7-2     °-7,? 

4S.70    °-^: 

0.S5 

(•48 

0.82 

1.05 

50 

00.11     °-^' 

,    20:45     °-7-' 

4s.ll     °-^^ 

O.sl 

0,48 

0.83 

1.07 

o.^l 

1    -   •'       0.7J 

0-37 

0.06 

0.  i<; 

(10 

50.30 

1    30.17 

4S.07 

11.72 

0.30 

0.77 

I)  40 

—0.83 

,—  1.09 

TIIK   OK  II  IT   OF    U  KAN  US. 


205 


TAItl.R   X,  AlUJ.  n.— Continued.                                                             1 

Arff. 

(I'.c.oj  Diir. 

(i).«.i)  uiir. 

(      ' )  Diir.  (f.N.a)  Diir. 

((•..•.2)  Diir. 

(...«.3) 

(u.cM) 

(.-.8.4) 

It 

If 

tt            II 

/»             /' 

"       II    j   //      " 

II          It 

tf 

If 

no 

r.9.;!0      „ 

80.17  , 

^;:2-°n  l:;^o.o. 

^;Jl+°- 

0.77 

0.49 

—0.83 

—1.09 

ni 

r.s.49-°-^' 

:ii(.H,H+°-7' 

0.73 

0..50 

0.84 

1.10 

(\i 

r.7.07    °-'' 

31.59     °1' 

47.24     °-'-'    l.HO     °°'^ 

0.71     °'' 

u.i;9 

0.52 

0. 85 

111 

(\:i 

M.HC,     °'^' 

;;.....,      070 

40.79     °-'j    1.95     °°'J 

0.H9     °"' 

0.00 

0.51 

0.87 

1.13 

i;4 

50.05     °'^' 
o.Si 

32:97      If^ 
0.67 

40.31     °-''''    2.05     °;° 
0.50               0.1 1 

^•"«  :.\] 

0.02 

0.5(; 

0.8S 

1.14 

05 

r>5.'_>4 

3.3.04 

45.81                2.10  , 

7.23  ,       , 

0.59 

0.59 

—0.89 

—  1.15 

(',(! 

54.44-°^° 

3..30+°^''' 

4-,  .,i-°S.l   .,  .,s+°'^ 

7.39+°;' 

0.55 

0.02 

0.91 

l.KS 

fi7 
(is 

5:!..;4     °-° 

31.95     °'5 
:5.,5S     °^ 

Jiiis  °P  0;  s^ii 

7.09     °     = 

0.53 
0.50 

0.05 
0.08 

0.93 
0.95 

1.10 
1.17 

f.9 

52.05     °    '^ 
0.79 

<»■'■«  IZ  ■"'  -s 

^•«^     o.\] 

0.47 

0.71 

0.97 

I.lH 

70 

51.2fi 

30.82  . 

42.93         ,      2.85  .      ,, 

T-f^^L^  ,, 

0.45 

0.75 

—0.99 

-1.19 

71 

50.47-°- J'' 

.37,41+°" 

4..  .>H-°^i    •Mc.+°  '7 

•^•"^"^o' 

0.13 

0.78 

1.00 

1.19 

7  J 
7:» 

li?,;;  - 

.37.98     °' 
3H.53     °--''' 

.8,20  °': 

H.31      °" 

0.41 

0.39 

0.H2 

0.S7 

1.02 
1.04 

1.19 
1.18 

71 

'••■'  :t 

39.00     °-2^ 
0.  ,0 

^"-'^^  ::;i  ='.^^"  ::;:; 

«•"• ::::] 

0.38 

0.91 

1.00 

1.18 

75 

47.43 

39.50 

39.51                3,75  , 

'^•■*!' .     0 

0.37 

0.90 

—1.08 

—  1.18 

70 

4(;.(;o-°  74 

4(1.05+°  ■•'' 

3s,77-°-7«    3,90 +°-'' 

8.57+°-°'' 

0.30 

1.00 

1.10 

1.17 

77 

4:.. or,    °7' 

40,52     °t7 

3S,01      °7"    .j.K;     °-'° 

8.01     °°7 

0.30 

1.05 

111 

1.10 

7S 

4.-..  2  4     °7' 

40,90     °l> 

37,23     °T^   4,37     °-' 

8.09     °°-=! 

O..30 

1.10 

1.12 

1.15 

7 'J 

44.52     °7^ 
0.70 

41.38     °-l^ 
0,40 

30.41     °T>   4.59     °-' 
o.Si  ■              o.Ji 

8.73     °°4 

O.OJ 

0.30 

1.14 

1.13 

1.13 

HO 

43.83 

41.78 

35.03         „      4,80  , 

8.70  , 

0.37 

1.18 

—1.14 

—1.12 

Kl 

4:1.14-°^'^ 

42,10+°  '« 

31,81 -°'^-    5,02+°  " 

H.77+°°' 

0.38 

1.23 

1.15 

1.10 

S3 

42.40     °^'? 

41.  HO  °^;^ 

42,51      °.'5 

33,98     °^^5,2t     °" 

8.78+°°' 

0.40 

1.28 

1.10 

1.08 

h:! 

42.83     °-'- 

33,13     °'^>    5,17     °-'-^ 

8.77-°°' 

0.11 

1.32 

1.17 

1.00 

H4 

41.15     °f' 

43.12     °-7 
0.  26 

32.28     °-^;    5.09     °-'' 
0..SO                0.22 

8,75     °°^ 
0.05 

0.43 

1.30 

1.17 

1.04 

H5 

40.52 

43,38  , 

31,12        0     5.91  , 

8.72 

0.10 

1.41 

—  I.IH 

—  1.02 

HI! 

.■!;i.90-°f^ 
.•i;t.:!0    °'*'' 

4:!,01+°■-^' 

30..55-°«7    ,i.l:{+°-'^ 

8.08 -°°,t 
8.02     °°'^' 

0.19 

1.15 

I.IH 

1.01 

X7 

4.3.82     °-' 

29.(18     °-^7    ,;.:i5     °-'^ 

0.52 

1.49 

I.lH 

0,99 

«S 

.M.Tl     "^''^ 

41.00     °  ''^ 

2H.H0     °f   0,57     °-''' 

S.55     °-°7 

0.55 

1.52 

1.17 

0,97 

SO 

38.14  °:i 

4  4.10     °"^ 
0.12 

27.91      °'^''   0.79     °-- 
o.yoi               0.21 

8.47     °°'' 

0.0(> 

0.59 

1.50 

1   10 

0.95 

00 

:!7.58 

41.28  . 

27. ni       „  17.00  , 

8.38 

0.02 

1.50 

—1.10 

—0.94 

ill 

.•!7.04-°-5> 

41,38+°'° 

20.12-°«''    7.21 +°'^' 

8.27-°" 

0.00 

1.01 

1.15 

0.92 

!)2 

:!:.52    °'^' 

41,14     °°^ 

25.23     °'^''    7.11      °;° 

8.15      °'-' 

0.71 

1.01 

1.14 

0.91 

!i;j 

;!(;.02    °=^° 

41,l8+°°-« 

21.33     °''°    7,00     °-    '^ 

8,02     °'-' 

0.7(> 

l.OC, 

1.13 

0.90 

y» 

0.  .(6 

44.48     °°° 

—0.02 

23,13     °''°    7,79     °    2 
o.Sy                  0.  IS 

7.89     °'} 
o'5 

O.SO 

l.OH 

1.12 

0.89 

!)5 

.'tr>.07 

41.40           , 

22,51         „    17,97  , 

7.71          . 

0.H5 

1.70 

—1.11 

—0.88 

!)(! 

;!j.02-°'3 

41,10-°  °^' 

.,,  ,;-,-o,80    >,  1,4-0.17 

in  °f  h:3,1  °  -^ 

7.58-°"' 

0.90 

1.71 

1.10 

0.H7 

!»7 

:!1,20     °  »* 

41.32     °°'' 

7.11  °;^ 

0.95 

1.72 

l.OH 

O.H(i 

its 

xuo   °-'i 

41.20     °'^ 

19.90     °i^    H.45      °'' 

7,21     °'7 

1.00 

1.72 

1.00 

0.H5 

09 

-■"  ::;;■; 

^'•"^  :::7' 

-■'  ::';;;;«"o  i:\i 

7,05     °"^ 
'          0,20 

1.05 

1.73 

1.01 

0.84 

100 

:!:?.(»! 

43,90 

H.IO         a.    8.73  .  „  ,, 

•■'^•"'     „  „ 

MO 

1.72 

—  1.03 

— 0.H3 

tol 

:i2.70-°^* 

43.70-°-'° 

17.31-°^-^  H.s.;+°-  ■ 

—0,  20 

1.15 

1.72 

1.02 

0.H3 

102 

.T.'.:t8    "•'' 

:;;:;! "" 

10.10     °''^    H,97     °" 

...     0,21 
(11 

1.21 

1.70 

1.(10 

0.H3 

lO.'l 

.-12.08     °'° 

1.-..02     °  ■'»   9.00     °°'' 

.   ...     0.21 
0.23 

1.20 

1.09 

0.9H 

o.H:i 

104 

=">^«     :J6 

42.92     °-"' 

•^•-     ::«;    '■•  '-^     Vol 

"•••'     III 

1.31 

1.07 

0.90 

0.83 

105 

ni.54 

42,01 

13.99                9.23  ,  „     .   5.79 

1..35 

1.05 

—0.95 

-0.83 

IOC, 

107 
los 

-m1 

,.j.,„-°-7')    .,  •!0+°°^    5.57~°" 
■    12     °  ^'^   ;     -      °°-^    534     °'^ 

?:    ° 7^  -9  °°! ':  ^  :'\ 

1.40 
1.44 
1.48 

1.03 
1.00 
1.50 

0.94 
0.92 
0.90 

(t.84 
OH,") 
O.HC, 

109 

^■•-^  oi; 

■'•■"^  :;?]  "'^  :::V'^^  -3 

1.53 

1.53 

O.SO 

0.87 

no 

30.03 

40.  r,  I 

10.19                9.13  ,             4.01 

1..50 

1.49 

— 0.8H 

—0.88 

111 

;to.5l-°" 

40.l7-°-'7 

,,  ,,,-0.70    ,,  ,,+0.01    ^    ,1,-0  2.» 

1.59 

1.45 

0.87 

0.89 

113 

:io.43    °°'> 

39,08     °   f 

H.SO     °'!'>    9.42-°°-    4.17     °  -'■' 

1.02 

1.41 

O.HO 

0.91 

It:} 

.•!0.:!5     °°7 

39.10     °-'i' 

HII     °'j'^   9.10     °°' 

3.91      °-' 

1.05 

1.37 

0.80 

0.93 

114 

;jo.;to    °°5 

3H.01     °5? 

7. .50     °  ^'*   9.30     °°4 

3.71     °-' 

1.07 

1.32 

0.85 

0.95 

— O.OJ 

0.56 

0.62  1              005  1              o-  23 

il5 

.10.28 

3H.fl5          _ 
.37.17-°S« 

0.S8         ,      9.31                3.48 

1.70 

1.27 

—0.84 

—0.90 

lir. 

;io.2n+°°' 

C,  .)s— 0  '•o    <i  .>,-o  07 

3.20-°-" 

1.72 

1.22 

0.84 

(I.!t7 

117 

:to.:i2    °°' 

3.;,.Hi;     °-^" 

5.71      °-^7    9  17     °°7 

3,01      °-^ 

1.73 

1.17 

OS  4 

0.99 

118 

.•to.;j7    °°^ 

30,21     °^' 
35,00     °^'t 

5.17     °-5-*   9.08     °°''    2.S2     °  -^ 

1.74 

1.12 

0.H4 

1,01 

119 

30.10     °"'^ 

4.05     °S-    8.99     °°'^    2.01     °-' 

1.74 

1.07 

0.H4 

1,02 

O.IO 

0,65 

0..49                0.  II                0.20 

120 

.30.  .^0 

34.95 

4.10               8.88               2.41 

1.74 

1.02 

— 0.H4 

—1.04 

25G 


T II  K   O  11  It  1  T   O  F    U  K  A  SVA. 


T.VIlLK  X,  .\[ui.  3,— ^»/l^/lH.7/. 

1 

Arg. 

(y.c.O)    Diir. 

,(».».l)    Diir. 

(o.i'.l;    Diir. 

i».«.2;Diir.  (.u.c.aiDill 

(('.8.3)  i|).r.3 

(i'.».4 

1       "      1 

"                    »f 

»/                      #/ 

1)         II 

»r                 »» 

"               n 

It 

130 

30. 5C  , 

3;;,5,s    °'"^ 

4.10 

H.NS 

2.41  ■ 

1.74 

1.02 

—0.84  l_  1.0 1  1 

iL'l 

ii0.70+°'» 

3.(;9-°'<7 

8.70-°'» 

2.31-°-^° 

1.74 

0.97 

0.8-1 

l.OC 

133 

30.,s5     °'-^, 

3.2(1     °-'» 

,S.03      °'-' 

2.02     °"^ 

1.73 

0.91 

O.8.- 

1.08 

i^a 

81.03     °  ''^ 

33.  ss    °-  ° 

2.S5     °-',', 

8.49     °''» 

1.84     °''^ 

1.73 

0.^(1 

0.8( 

1.09 

124 

31.33     °-'° 

33.15     °-   • 

a. 47   °'!; 

8.34     °:5 

1.C7    °''; 

1.71 

0.81 

0.87 

l.lu 

o^.} 

0-7J 

0.34 

0.  tO 

0.  i() 

li'S 

31.4(1  . 

•■"•^•■2     „  ,. 

2.13 

S.18             , 
s.<.2-°"* 

1.51 

1.70 

'•■  0.77 

—0.88 

_1.11 

llT. 

31. 71+°='-^ 

30.C7-°!-! 

l.«l-°-'-' 

l.:io-°;5 

l.n7 

0.72 

0.9( 

'     1.13 

137 
13S 

81. lis     °-7 
3.)  0^     °-.lo 

30.1.1     °lt 

i..5a  °-;;'  7.S5  "■;' 

1.30     °-   ^7.07     °-   •' 

1.21    ° '3 

1.08     °'' 

1.05 
1.(13 

0.C8 

o.(;3 

0.93 

1     1.13 
1.13 

139 

33!  (Tl     °-'3 

3«.39     °1' 

1.03     °:''-7.48     °"> 

0.90     °''- 

1.59 

0.59 

0.94 

1.14 

°-35 

o.-.S 

0.30 

0.19 

0. 12 

I.'IO 

33.!)fi  ,        _ 

!    27.00     „  , 

0.S3          , 

7.29     „ 

0.84 

1.5(1 

0.55 

0.95 

—  1.15 

l.'U 

s:i.:n+°-3^ 

3C.s3-°-   " 

7  09-°° 

0.74 -°''° 

1.53 

0.51 

0.9(1 

1.15 

1.13 

33.74     °f 

2C.03     °-." 

"••*  n^ 

(;.,S8    °- 

0.05     °°'' 

1.49 

0.48 

0.98 

1.15 

l:t3 

l:!4 

31.1(1     °-»- 
3  4.  CO     °-«/ 

"5  -'l         "'J 

24:i]  If 

0.44  °"° 

0.40     °-' 

:.i'i " 

1.45 
1.41 

0.45 
0.43 

0.99 
1.01 

1.15 
1.10 

0.4O 

0.79 

0.04 

O.-M 

0.05 

l,!.') 

35.0(i  ,  „    „ 
35.54 +°-»'^ 

23.  C5 

0.33 

0.25 

O.40 

1.37 

0.40 

—1.03 

— l.Ifl 

mr, 

33.H(J-°-7y 

0.32-°°' 

0.03-°- --• 

0.41 -°°5 

1.33 

0.38 

1.05 

1.15 

l:!7 

8C.05     °5' 

33.0(5     °''° 

0.:{5+°°3 

5.SI     °-- 

0.38     °°.? 

1.38 

0.37 

l.OC 

1.14 

i;)s 

3C..5S     °-5' 

31.37     °'" 

0.40     °°5!5.5»     ° -- 

0.30-°-°^ 

1.33 

0.35 

1.07 

1.13 

l:i<.) 

37.13     °55 

20.4S     °-7'^ 

0.49     °°'^  15.38     °-' 

O.30     °°° 

1.18 

0.34 

1.08 

1.12 

°-57 

0.79 

0.12 

0.  J2 

0.00 

140 

37.70  ,      , 

10.00 

O.OI  , 

5.10 

0.30  , 

1.13 

0.34 

—1.09 

—1.11 

141 

3s.:io+°-^° 

H.'iO— °''' 

0.70+°';' 

4.95-^-' 

0.38+°°' 

1.09 

0.33 

1.09 

1.09 

143 

3><.ii3     °-^'- 

ls.13     °"7 

0.04     °'" 

4.73     °-- 

0.41     °°3 

1.01 

0.31 

i.lo 

1.07 

143 

3'.t.55     "l'' 

17.30     °'J 

1.15     °-^' 

4.52     °'-' 

0.45     °°» 

1.00 

0.31 

1.11 

1.00 

144 

40.31     °-^''' 
0.O7 

IC.fiO     °  ''' 
°-75 

1.39     °  -^'   4.31     °-' 
0.J7               0.21 

0.50     °°-^ 
C.07 

0.95 

0.35 

1.13 

1.05 

145 

«>•■'*'*  ,      ^0 

15.,S5 

1.00  . 

4.10 

0.57 

0.91 

0.30 

—1.13 

—1.04 

it<; 

41.5(J+°-<iS 

15.12-°'' 

1.93+°- ^9 

3.90-°-° 

0.05 +°°« 
0.73    °°** 

0.8C 

0.3S 

1.13 

1.03 

147 

43.37     °"' 

U.;i9     °" 

2  OS     °.?.'> 

3.71  °;'^ 

0.f<3 

0.40 

1.13 

1  03 

14V( 

43.00     °"-' 

13.07     °-   -' 

2.  ("3     °-'i 

3.53  ° "-' 

0.82     °°? 

0.7S 

0.41 

1.13 

1.01 

1,9 

43.73     °-7^ 

13.97     °J° 

3.01     °-''^ 

3.33  °''' 

0.93     °" 

0.74 

0.44 

1.14 

0.99 

0.76 

0. 69 

0.41 

0.  IS 

0.  II 

150 

44.51  , 

13.2S        ,„ 

3.42  , 

3.15 

1.04 

0.71 

0.4C 

—1.14 

—0.98 

151 

45.;io+°" 

ii.co-°ff 

3  8C+°-'/ 

3. 98-°- 'J 

1.10+°'^ 

o.(;7 

0.50 

1.14 

0.9C 

153 

4C.0'J      °','^ 

10.04     °-f 

4.33     °-^^ 

3.S2     °'^' 

1.29     °-   •' 

0.C4 

0.53 

1.13 

0.95 

15;) 

4C.-I0     °-^' 

10.30     °-^* 

o.(;8    °f- 
0.60 

4.,S0     °f 

3.00     °"^ 

1.43    °;\t 

o,(;3 

0.5C 

1.13 

0.94 

154 

47.73     °i- 
o.S.( 

5.29     °-''^ 
°  5j 

•■'•^i  o::i 

»-S^     0:;^ 

0.59 

0.59 

1.13 

0. 93 

155 

•*•'*  •''f  ,      Q 

9.  OS 

6.S2  , 

2.37 

1.74 

0.57 

0.03 

—1.11 

—0.92 

15(; 

4!).43+°'^-^ 

S.49-°-^? 

o.3c+°-  ;» 

0.92     °'^'' 

2.35-°'^    1.90+°-'°  1 

0.55 

O.C(! 

1.10 

0.91 

157 

5(..39     °-^J 

7.93     °" 

3.13     °'- 

3.07     °'l 

0.53 

0.70 

1  09 

0.90 

15H 

51.17     °'^^ 

7..i7     11^ 

^••^l     V,1 

3.01     °'^ 

.>ot     °'7 

0.53 

0.71 

1.08 

0.89 

159 

53.07     °'-'° 

'•-•«*     «    •' 

8.12     °V^'    1.91     °'°l 

■^■'^■i  °!'^ 

0.51 

0.78 

1.00 

0.88 

0.90 

0.50 

o.(J.' 

0.09 

0.  I.S 

ICO 
ICl 
ir,-2 
li;3 

53.07  , 
5:(.H.|+o-9^ 
54.S3    °W 
55.75     °y.5 

"■s5-°-'' 

0.40 

J-::^  COS 

1.  i4 0 

2  97     °"> 

3  1(      °"' 

0.20 

0.50 
0.5(» 
0.50 
0.5(1 
0.51 

0.82 
O.KC 
0.90 
0.91 
0.97 

—1.05 
1.04 
1.03 
1.02 
1.00 

—0.87 
0..S7 
0.87 
0..80 
0.80 

ii;5 

57. CO  , 

4.14 

12.10 

1.51 

3.55 , 

0.52 

1.01 

-0.99 

—0.80 

ICC. 

5.s.C3+°'^' 

3.77 -°-37 

12.«0+°-7° 

l.48-°°.5 

3.75+°-*° 

0.53 

1.05 

0  97 

0.87 

Km 

5'...C0     °-'^^ 

3.42     °''=' 

13.53     °'- 

1.4G     °°^ 

3.94     °"' 

0.54 

1.08 

0.90 

0.87 

ics 

C().5S     °-''^ 

3.10     °-^-\ 

14.25     °-7.'    1.45-°°' 1 

4.14     °-^° 

0.5C 

1.11 

0.95 

0.88 

IC'J 

C1.57  y> 

3.M0     °-5° 

15.00     °'5 

1.45     °°° 

4.34     °'^° 

0.57 

1.15 

0.94 

0.88 

°-<)') 

o.iS 

0.74 

+0.01 

0.19 

i 

170 

r,-2M, 

0  -,•> 

15.74  , 

1.40  , 

4.53  , 

0.59 

1.18    '• 

-0.93 

—0.89 

171 

C.-i.5(!  +  '°° 

n;,:— 0.25 

lC.49+°-75 

1.48+°" 

4  7.,+°>9 

0.(;3 

1.21 

0.93 

0.91 

173 

C4.5C      I"" 

37(5    °" 

17.25     °'''    1.51     °°-' 

4;, II      o.") 

0.C5 

1.33 

0.91 

0.92 

173 

fi5.57     "•°' 

1.S4     °-' 

1.S.03      °'J'ir,.^      °°4 

5.09     °''~^ 

0.C7 

1.25 

0.90 

0.93 

174 

(iC.58     '•°' 

I.OJ 

'■"^  0:;^ 

.8.78  °:;; 

1.00    °-°? 
o.oO 

•^-^     0::;^ 

0.70 

1.37 

0.89 

0.94 

175 

C7.C0 

I  51 

19.55  ,        0 

•••'fi  .     «, 

5.40,        ^ 

0.72 

1.30    ■ 

-0.89  . 

—0.95 

17(! 

CM.CI+'°' 

1.3.S-°'' 

20.:!3+°-7^ 
31.09     °'^ 

I  -.,+007 

!-'^>      COS 
1..V9     °°'^ 

5.04+°- ''^ 

0.75 

1.30 

0.88 

0.90 

177 

(•.1».C3     '•°- 

1.27     °'i 

5.81     °'J 

0.78 

1.31 

0.88 

0.97 

I7H  1 

70.C5     '■°' 

1.19     °°'^ 

31.80     °" 

5.97     °'^ 

0.81 

1.33 

0.88 

0.98 

17'.t 

71.07     '•°-' 

1.13     °°^' 

32.03     °-!7 

'•"^     °?^ 

0.13     °'5 

0.84 

1.33 

0.88 

0.09 

1.02 

o.o. 

0-77 

0. 10 

0'^ 

1 

ISO 

72.09 

i.os        '  1 

23.40 

2  08 

C.37 

0  S8   ' 

1.34 

-0.S8  _1.00  1 

T  U  K   O  U  IJ 1  T   O  F    U  11 A  N  U  S. 


267 


TAIII.K  X,  Ami.  y,.—V<mlhiH,,l. 

Arg. 

(w.c.O)    Did. 

(.u.d.ij   Diir. 

(u.f.l)    DilT. 

(i'«.2)  Diir. 

(0.0.2)  Diir. 

(u.*.a) 

,1' 

{v.c.ii)  (i),«.4J 

//       1       tt 

r " 

n              n 

»       // 

It          It 

"                    // 

IHU 

72.(!» 

1.08 

2.1.40  , 

2.08  , 

(>.27  . 

0.88 

1.34 

'_0.8S 

—  1.00 

isl 

.1.71+'°' 

1.077°°' 

21.17+°" 

.,  .>0+o'J 

(l.42+°-'^ 

0.91 

1.31 

O.MS 

1.02 

18i 

74.7a     '°'' 

1.08+°°' 

21.92     °-7:. 

2.;'jl    °" 

0.55     °'' 

11.91 

1.34 

0.88 

1     1.03 

lb.i 

7.-..7r,    ••»■' 

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2.i8 


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1,37 

1.14 

0,  !I8 

0.  !(5 

'.'■:7 

IdO.HT     '^  5'J 

27.05     °-°° 

1    32,  S  J     °-7 

a.oo    °°'  4,70    °-°p 

1.3S 

1.12 

0.08 

0.00 

•jss 

luo.27     °''" 

27.04-°°' 

32.. 55     °^7 

a.OS     °°'    4..s;-l      °'°'* 

1.3S 

I.O!l 

0.07 

0.it7 

:!s>.( 

'J'J.07     °-?° 
0.61 

27.03     °°' 
0.01 

32.2'J     °='^'    3.07-°°'    4.00     °°.'; 
0.27  1              0,00                O.OS 

1.30 

1.07 

0,00 

0.!»7 

i'!tO 

HD.Ofi 

27.02 

1    32.02             '3,07              4.05 

1.30 

1.05 

—0,00 

— O.ilS 

•Jill 

Its. .15—°'" 

27.00 -°'°- 

31. 70-°--^' 

.i.r,7    °°°  .V(M)+°-°; 

1    (0 

1.02 

0,!IO 

((.!IS 

'.'!l-i 

!i7..s:i   °''- 

27. .5S     "•°- 

31.50     °-^^ 

0.00    ..,,„,      0,0^, 

1.10 

l.flO 

0,00 

0.00 

■J'.i:i 

!    .,.._,,      0.62 

2750     °°' 

31,23     °V7    .•t.,J7     °°°   5,11      °^.^ 

1.1(1 

0.!is 

0. 05 

1.00 

L".i| 

-..(i.O.S     °^' 

...    .  ,        O.OJ 

2  1  .  <i  4 

:    30.1.7     °-="3.08-^°°'    5,10     °°^ 

1  40 

0,05 

0.95 

1  01 

0.6} 

O.OJ 

0.26              0.01               0.05 

L'!tri 

(•.'■i.il.'i 

27  52 

1    30.71            ,    3.00               5.21   ,  _ 

1.30 

0,03 

—0.05 

—  1.02 

•J'.ii; 

".I.-,  ;jl-°'^'t 

27  ts-°-°-« 

30    5-°-='''   3  70+°°'    5,20+°'° 

l.3!l 

0.01 

0,05 

1  (12 

•-".•7 

i   '.•l'.(;7    °1'* 

27   15     °"-' 

.■„;,,    0.26  ,;  ,    0.0,  ,,,,    0.0, 

1  :;s 

O.ss 

o,!i(; 

1.02 

•.".IS 

'   '.M.ol    -  '.< 

27.12     °"-^ 

2'...oa    °-'^'  .•;.7J    °°'  •V37    °": 

1  :;s 

o.sr. 

0.00 

1  (i:: 

•J'.i'.i 

i.a.Ki    °'' 

27., S     °-°> 

2o.r,7  °-"^'  3.71  °°-:  r..i2  °';: 

1  :;7 

0.S4 

0,07 

1.03 

0.(1  { 

0.0. 1 

0,27 

O.UJ                        u  u^ 

:iiio 

,    !f-'.77 

1  27. a  1 

20. 10 

3,70              5.17 

1  35 

0.S2 

—0,07 

1.03 

TlIK   O  II  HIT   OF   UUANUS. 


2i>i) 


T.VIil.l-:  X,  .Villi   •.i.-^-r„i,in,ii,d,                                                    1 

Arg. 

(v.r.O)   Diir. 

(r.r.l)     Itiir. 

(.■>.2J  HilV. 

v.i:i)  \)\fS.  (U.t.-i) 

(D.c.a) 

»t 

(v.*A)  (U.C.4) 

II     \     II 

M                  It 

•• 

tt 

:jou 

l.-...l;|-°f» 

27.  :u 

20.40 

a.7rt  ,        '5.47  . 

1.35 

0.82 

—0.97 

—  l.l.l 

;iiil 

•.'7.:io-°°» 

20. 1.1 -"■»•» 

.       +0.0.  ,,.,,+0.04 

1.3» 

0.80 

0.97 

1.03 

:iO'j 

1.1. :.o    °'-» 

•-•7,-.'.l     °°^ 

2H..ia   °-'^' 

3.80     °°«  5.50     °°5 

i.aa 

0.78 

0.98 

1.03 

:iii:i 

1M»..>.7     °'' 

27.21     °° 

a.s-j    °'°-'  :,..ii    °°5 

l.ai 

0.70 

0. 98 

1.04 

Mi 

o.0.( 

27. 1«     °°5 
0.06 

^«-'"     ajo 

a.85    °°]  5.00    °-° 

o.oi  J             0.05 

1.30 

0.74 

0.90 

1.U4 

:io.'< 

SI). CO         - 

27.10 

2S.11 

3.87  ,          15.71  , 

1.28 

0.72 

—0.99 

—  1.04 

.•W 

MS.!.;  -°f< 

27.0,^ -°°5 

27.Ml-°'=' 

a.01+°°»  5.75+°°' 

1.27 

0.71 

1.00 

1.01 

•.m 

M.r,    °'^ 

2.;.!.!)     °°' 

27.. 10     °-"' 

3.04     °°'   5.80     °°5 

1.25 

0..1U 

l.lll 

1.(11 

;!(is 

.S7.7U     °'-' 

2.i.i.:»    °°^' 

2  .a:,    °-'^ 

3  07     °°'   5.S4     °-°-» 

1.23 

0.08 

l.ll 

1.(11 

M',) 

■^^'" ::::; 

2.;.«.i    °° 

0.07 

^^'« ::;; 

4.01  °°;  5.8S  °'°; 

0.04              0.04 

1.21 

u.oo 

1.01 

l.UI 

•Mo 

fH.I!)         , 

2(1. 7  9 

2(!.S.'i           , 

4. 15  ,            5.93  , 

1.19 

0.05 

—  1..I2 

—  I.OI 

:ill 

s.-...si.-^-  '° 

2.i.72-°° 

2(i.r.!.-°-"^ 

4...0+°°»  5.1.r,+°°-» 

1.17 

0.04 

1.02 

1.(11 

:!l'i 

K,.-><>     °   '° 

2.1.  C,.^     °'" 

2(1.  a  4    °-'5 

4.14     °°^   .i...O     °°' 

1.15 

U..13 

l..ia 

1.(13 

:i  1  ;i 

Mi.r.i.    °  '^ 

2r,.:,8    °°7 

2.1.0-      °-'-^ 

4.1s    °'°'  o...a    °"' 

1.12 

0.12 

1  .13 

l.(i:i 

lilt 

o.j.S 

^"•^u    °°'^ 

2.-..M  °---: 

4.22     °"'  0.07     °-°* 

1.10 

U.02 

1.01 

1.03 

0.09 

0.J5 

0.05                 O.OJ 

ai:. 

>*:i.M 

2(!.40 

2:..  59 

4.27  ,          'rt.IO 

I.  OS 

0.01 

-L.ll 

—  l.O'l 

:ilA 

Mj.!.a-°5'^ 

21;. ai  -°°'' 

2.-..ai-°-> 

4.32+°-°.''    .l.l(+°°t 

1.05 

0..1I 

1.11 

1.(12 

lin 

2(i.22     °°'' 

2.-..  10     °-l 

4.37     °'°-^    .1.17     °-°-' 

1.03 

0..1I 

l.ll 

1.(12 

:ils 

2ii.l2     °  '° 

2t.s5     °-5 

4.43     °°''   0.10     °°- 

l.lll 

0.01 

l.ol 

1.01 

:ilu 

81.27     °-^' 
0.54 

2(1.02     °  '° 
0.  u 

24.00     °  ''^ 

4.48     °°-=i,0.22     °°' 
0.00  1              0.0  I 

o.us 

0.01 

I.IM 

1.00 

:!'.'o 

HO.  7.1 

2r..!»l 

2l.n5 

4.51  ,        /0.25 

0.9.1 

0..11 

—  l.ot 

—1.00 

.■!2l 

."^ll.-.'K— °-.'!' 

2:...so— °" 

21.11-° -'» 

1  i;(i+°°^'  0.27+°-°' 

0.91 

U.OI 

lot 

0.99 

:i-2i 

7:m;s    °-^- 

2:..  (IS     °'' 

2a.  H7     °-' 

4.(1.1     °°?  0.20    °°- 

0.91 

0.02 

1.113 

0.99 

-.ij-.i 

71M7     ''•^' 

2.-..5(i     °" 

2a.. la   °»> 

4.72     °°f   0.30     °'°' 

0  89 

0.02 

1.0.: 

0.98 

■.i-2i 

78..iS     ^-''^ 

2:..u    °'' 

2a. 39   °-'t 

4. 78     °°^':«.a2     °°- 

0.87 

U..13 

1.03 

0.98 

0.49 

O'J 

°-'3 

0.07 

0.02 

:ij:. 

7S  11) 

2-..ai 

2a.  in 

4.S.1  , 

n.ai 

0.S5 

0.04 

_l.o:' 

—0.97 

■iJi; 

77.7:! -°-'J 

2,-,.l7-°'t 

•  .)  .,..-^  •!4 

,  .,..+007 

0.35+°°' 

0.83 

0.05 

1.03 

0.97 

;!i'7 

77.-.(:     °"^ 

2.-.. 02     °'-^ 

2i''.ii9     °-' 

TOO     °°7 

0.35   °°° 

0.8l 

0.00 

1.02 

0.90 

.•;js 

7r,.s-j    °t» 

21. M7     °  '.i 

22.4.!     °-' 

5,.H1     °°7'o.35     °°° 

0.79 

0..17 

1.02 

0.90 

;{;!'.» 

7.i.;jlt     °'' 
0.4J 

24.72     °';' 
0.  lO 

22.23     °-' 
0.  ij 

,-.,13     °°7    ,i.:,,-i     0.00 
0.07                0.00 

0.77 

0.(19 

l.ul 

0.96 

;j:io 

7r..tt7 

2l..'>il 

22.00 

5.20  ,            n.35 

0.75 

0.70 

_1.01 

—0.95 

:i:n 

7.-..;,7-°-'° 

2i.;!i.-°-;7 

21.7S-°" 

5.27+°°7'0.;!5     °'°° 

0.73 

0.72 

1.00 

0.95 

;):!2 

7:.. IS    °-''^ 

21.i'l     °-\l 

21.57     °-^' 

.-..34     °°7    0.31-°°' 

0.72 

0.74 

1.00 

0.95 

:(:!:( 

7  LSI     °V 

2i.oa    °''' 

21.  a5    °-'-' 

5.41      °°7    ,!,:j:,     o°' 

0.70 

0.70 

0.99 

0.95 

.'i;i  1 

7...  -!^ 

2a.  S4     °"^ 
0.  ly 

21.14     °'' 

0.20 

5.4S     °-°7    0.32     °°' 
o.on                0.02 

0.09 

0.78 

0.99 

0.95 

3.1.'i 

71.11 

2a.(i.-. 

20.01 

5.:.(!  ,             «.30 

0..1S 

O.so 

_0.91l 

—0.95 

.'):;i', 

-.:;.71.-°-'-' 

2:i.4.-,-°-  ° 

20.73-°-" 

.•,.(13+°°'    .I.2S-°-°' 

0..17 

0.S2 

0.9s 

0.95 

;i:J7 

7:!.  IS     °-" 

-'••!-•»     °' 

20.53     °-'° 

5.70     °°7    .1.25     °-°-' 

0.(1(1 

0.s| 

0.97 

0.115 

;!;!■< 

7.1.  IS     °-^° 

2.i.02     °- 

2...  a  4     °"' 

5.7s     °°'^   0.22     °-°' 

0..15 

O.sil 

0.9.1 

O.li^i 

HT 

U.'A     °'l 
0.  J, 

'■'■''  o:;i 

20.15     °    " 
0. 19 

5.85     °°7    0.19     °°-5 
0.07                0.04 

0.04 

0.88 

0.95 

0. 95 

:u(> 

7--'.r,(! 

22.. 'a 

1^5"''         r.     ., 

5.92  ,        ,    0.15 
5.1.8+°°'^'   .1.11-°°' 

0.01 

0.91 

_0.1I5 

—0.95 

::n 

7J.lJ-°-'-» 

22.aa-°;} 

l'.».70-°' 

o..;:i 

o.na 

U.lllt 

0.9(1 

:;  1 J 

72.-'o    °-' 

22.00    °-  j 

10..  2     °' 

(1  (.5     °°7    ,;..,7     °-°-' 

0..i:i 

0.1.5 

11. 11 1 

0.9(1 

.'ill 

7J.no    °-  ° 

-;  li 

10.45     °     ? 

0.12  =°  .1...2  °°-: 

.1   10     °°'    -.  '17      °°-^ 
''•'•'      0.06    ••   ■''       0.00 

0  11:! 
0..13 

(Ills 

1.00 

0.11 » 

u.  ii:; 

0.11(1 
0.97 

:ti:. 

7l.r,(! 

21.  a  I 

1  0. 1 .-) 

r.  2.'>         ,15.01 

0.03 

1.02 

_0.03 

0.97 

.'Mi; 

7l.r.i-°'t 

21.(11  -°-7 

l;i.o.i-°  '5 

,.  .,,4.0.06   r,s,;-0.05 

0.04 

1.05 

0.1.3 

O.lis 

:ii7 

71.:!!t    °'' 

20.7.-i     °''> 

1S.S7     °  '.' 

0.37     °°''   5.S0     °-°''' 

0.04 

1.07 

0.114 

O.l'l. 

Ills 

7l.-.'!t    °'° 

2(M(l     °-''' 

ls.74     °'5 

0.43     ""^   5.73     °°' 

0. 05 

1.09 

0.94 

1.(111 

lii'.i 

71.21     °°,^ 
0.00 

20.10     °-'° 
0.31 

1.S..12     °'^ 
0.  lO 

0.48     °°5    5.00     °-°' 
0.05                0.07 

0.0.1 

1.12 

0.94 

l.dl 

n.Mi 

7l.l-< 

lo.s'i 

18.53 

0.53  ,            5.59 

0.07 

l.lt 

—(1.94 

—  1.02 

:!.-.l 

71.10-°°^ 

iii..-,.-°-^' 

IS.  43-°°'' 

0.58+°-°.=!  r..5i-°-°'' 

0.08 

1.10 

0.111 

1.(12 

:i.'>2 

7l.ltS-°°-' 

I1..22     °-5- 

is.  31      °°'> 

0.02     °°'   5.(3     °°''* 

0..11I 

1.18 

0.95 

1.03 

;i:..i 

7I.US     °°° 

Is.llO     °-'^ 

IS  ■..,..     °o« 

0.00    °°'  .-..a.-,    °°-] 

0.71 

1.20 

(1.95 

l.dl 

;!.')» 

71.10+°°' 
0.05 

IS.f,.!     °-" 
034 

l.s.Il)     °°7 

0.05 

0.70     °°'   5.27     °°'' 
0.05               0.09 

0.72 

1.22 

0.95 

1.05 

;i.v. 

7i.ir>  ,      , 

IS.  23 

IS. I  1 

0.73  ,            5.1s 

0.71 

1.23 

—0.95 

—  1.(10 

:i:.ii 

.,  ,,+0.06 

17.ss-°-.V» 

|s.|,|-°°-» 

.1.70 +°°»    .^..0-°-°'' 

0.70 

1.25 

0.1..1 

1.(1.1 

.•1.-.7 

71.-.t     °°'' 

n./ia   °-^; 

IS, 07     °°.'   .1.70     °°-'   •,.....     °°'J 

0.7s 

1.20 

0  97 

1  .1(1 

:i:.s 

71.40     °-'' 

17.17    °-''' 

ls.,|-,-°°-    r,.sl     °°-    4.1.1     °°'> 

0.8.1 

1.27 

...lis 

1  (I7 

.'!;.!i 

7i.r,-2    °'- 

Ki.so   °-." 

l.'i.05     °°°    .I.sa     °°-    4. SI      °'° 

0.S2 

1.2s 

.1.119 

1  (17 

O.I^ 

o-,?7 

+0.01  ,             0.01              0. 10 

;;(;o 

7i.i;7 

ir,.4a 

IS. on             |fi.S4               4.71 

O.sl 

1.29 

—0.911 

—  1.07 

260 


THE   OIIUIT   OF    UKA  NiTS. 


TAIILK  X,  AiKi.  ■.\.--i:<n,liiiiir<i.                                                             1 

Art?. 

(i).c.O)    Diff. 

It              It 

(v.».\)    Diir. 

(i).f.l)    Diff, 

(0.8.2)  Diff. 

(i'.c.2}  Diff.  (i).«.3)  {U.C.3 

If                **             1          "             j         " 

(v.hAj 

It 

(0.C.4) 

ff            ff 

tt               II 

" 

3C.0 

7l.f.7  , 

l.i.43 

1 8. or,  , 

(1.84 

4.  VI 

0.S4 

1.29 

—0.99 

-1.07 

;ti;i 

7 1.84+°- '7 

i(;.05-°-^ 

is.os+°°-' 

f..84     °°° 

4..;i-°'° 

0.87 

1.30 

{      I.UO 

1.07 

:ii;2 

7i>.03    °-"* 

l.-,..;7    °-''^ 

18.12      °°^ 

r..«4     °°° 

4..',l     °'° 

0.  8U 

1    1.31 

l.OI 

1     1.07 

3(;:j 

7-.'.'.'4     °-'' 

1.5.28     °y> 

18.17     °°: 

.1.84      °°° 

4.41     °'° 

0.91 

1.31 

1     1.02 

1.07 

•Mi 

0.25 

14.811  -"-^'^ 

0.00 

.  — 0.01 

4,31     °'° 
0. 10 

0.94 

,    1.32 

1.03 

1     '"^ 

;w:> 

7'-'.  73  , 

11.50 

18.32  , 

.1.81 

4.21 

0.90 

1    1.32 

—  1.03 

—  J. 07 

'MU\ 

7:!,(iO+°='7 

,,  ,,-o.,?o 

18.42+°'° 

.•,.79-°°=' 

-co 

0.98 

1.,'I2 

1.03 

1.0; 

3i;7 

7.i.30    °:f 

1,1.71     °{° 

1V.-.3     °," 

.1.7.;    °°-| 

4.01     °'° 

1.01 

1.32 

1.04 

1.0(1 

3(;s 

73.02     °-'- 

I3.:;i    °'° 

IS..,.:    °-  •' 

'•  73       •' 

.3.90     °" 

1  03 

1.31 

1,05 

!.().'> 

3('i!) 

73.9.;    °-'' 

12.91    °-^° 

'^•^'  :::^ 

.;:.;8  °°-^:3.8o  °'° 

1.00 

1.31 

1.0(1 

1.0.-. 

o-,?7 

0  ;o 

0.05  '                 O.IO 

370 
371 

!;?;«,•■; 

12  5' 
12I0-^-}" 

1^:?^^::^ 

(i.ns 

(•  -.s-""-^ 

'••^■^   0.06 
1.4. 

;;;^2-°-° 

1.08 
1.10 

1.30 

1.29 

Ll.07 
1.07 

—  1.0.5 

1.0.'. 

373 
373 

JJIS  •- 

IIS  -r 

0. 20 

'■'■••'■'  0.  = 

19. .-.7     : 

.1.41     ''•'° 

1   12 
1.14 

1.28 
1  -*7 

l.(t7 
1.08 

1.04 
1.04 

374 

*•*••'•'      0.46 

0.40 

0.07 

3.32     °°'^ 
o.oy 

1.1(1 

1.2,', 

1.08 

1.03 

375 

7l!.4.-> 

10., 50 

20.04 

.1.32 

,3.23 

1.18 

1.23 

—  1.08 

—  1.02 

371! 

7(1.94 +°-»'^ 

10.10-°-'° 

2o,:w+°--f; 

.1.24-°°'^ 

3.1:-.-°°S 

1.20 

1.22 

1.08 

1. 00 

377 

77.44     °-50 

9.71     ° -V) 

20.58     °-^'' 

.1.15     °°'J 

3.07     °°-^ 

1.22 

I.2.. 

l.OS 

0.1)1) 

37  S 

77.9fi     °-5- 

9.32     °  -''^ 

•)((  .V)      °  .^' 

(i.O.!     °°'J 

2  911     °°''* 

1.23 

1.19 

1.08 

0.98 

379 

78.,M     °- ■''•'' 
c-57 

8.93     °.W 
0  ,19 

21.21     "=•'- 
OJ4 

,V9.;     °"° 
0. 1 1 

192     °°7 

0.07 

1.25 

i.in 

1.08 

0.97 

380 

79.08  . 

8.54 

21.55 

,"l.S"l 

2.8.', 

1.2fi 

1.13 

—  l.OS 

—  0.9(1 

»,'» 

.,,,,,+0.58 

s.ic-o.?s    ...i.n|-'-^-.?f' 

5  74-°" 

2  78-°°" 

1.27 

I.tl 

1.07 

0.9.1 

3«3 

W...27     ^•^" 

7  7n     •^■M     .>•«  1)^     °  .W 

.•,.f.3     °" 

:.72     °°'' 

1.28 

1.09 

1.07 

«,  95 

3S3 

80. 9C     °-^'3 

7:43     °^'l     22:r,7     °'" 

.-..51      °'' 

2.(17     °°5 

I.2S 

\.W, 

1.07 

0.91 

384 

81.51     °   ••♦ 
0.(15 

7.07     ''^^' 

0.J4 

23.08     °-" 
0-4.} 

.V39     °'-' 
0. 1.' 

2.tl2     °°'' 
0.04 

1.29 

1.04 

1.01; 

0.0:1 

3S5 

Sl'.IO  ,      ,„ 

fi.73 

23.51 

5.27 

2.. 58 

1.29 

l.OI 

—  I.Ofi 

—0.92 

3Sfi 

8.'.87+°-^"^ 

.;.39-"^'-' 

2.1.9-,+°-»/ 

.-.  14-°'-^ 

.» r,,i— °°3 

1.29 

0.98 

I.O.; 

0.92 

387 

83.5.5     "^"^ 

<;.0fi    °-^-^ 

21.41      °-^'' 

.-,.01     °'.' 

•>  riii    °°^ 

1.29 

0.9(1 

l.o,-. 

0.92 

38  8 

84.^7     °'' 

.5.74     °-^-' 

•'i  sit     °-»'^ 

'■^^     oil 

o'Vl       °°- 

1,28 

0.93 

lot 

0.91 

389 

«^o'  c°;;i 

,5.42     ^ 

25:38     °-'"> 
05' 

^•'■•«  r.: 

2:Vj-°°- 

0.00 

1.28 

0.91 

1.03 

0.91 

3!tO 

85.7.'>  ,        , 

5. 1  2           „ 

25.89 

459 

2.49 

1  27 

0.R8 

—  1.02 

-0  9>» 

3!ll 

.V,.-,1+°-7^ 

4.84^-"' 

..,;  4.>+°  5> 

115-°" 

.»  x<\     °°° 

1  20, 

0.8fi 

1.01 

0.90 

'A'.l-2 
3!I3 

S7.29     °^^ 
88  OS     °" 

4. 30  °  f ; 

27.51      ^  "' 

4:11     °    j 

2:-.0+°°' 
.)  -,.>     °°- 

1.25 
1  23 

0.83 
0.81 

1.00 
0.99 

0.90 
0  90 

3!U 

««•««  :::s; 

^•""  l\\ 

28.0.S     "l'. 

^•"^     olj 

•155     °°-' 
0.05 

1  22 

0.78 

0.98 

0.90 

3M 
3ftfi 
3'.I7 
398 
3!)  9 

9.'.  39     l\i 

0. 1  ^ 

2s  (;7        ,     3  s7 

„,,  .,'. -{-of'o  ,,  -.5-°  il 

2    is     ^^'     is     °'5 
0.05              0.14 

2  58 

2..;3+°°^ 

2.(18   °°; 

2.74     °°'' 

"sl     °°V 

0.07 

1  ;>0 
1.17 
1.15 
113 
1.10 

0.7.1 
0.71 
0.72  ' 
0.70 

0..19  1 

—0.97 
0.9.-, 
0.91 
0.9:1 
0.92 

—  0.90 
0.91 
0.91 
>'.1'2 
0.92 

400 

94  02 

2  91 

31.79        ,,  3.10 

2  S8 

1  OS 

0.(^8 

—0.91 

-0.93 

401 

•11  !lo+°'^^ 

2.77-°'-' 

;i2.45+°''''  302-°'' 

._,,„;  -1-0.  O.'l 

1.05 

0.117 

0.91 

0.93 

40:3 

•I.-..  83     °"' 

2  •;5     °  ' ' 

..J  ,,    o.r.6  ^,  ^,,    O.I.? 

.i.mi     °  '° 

1.02 

(I.C.C 

0.90 

0.94 

403 

9l;.7.^    "■'■'-■ 

"..',(!     °°'^ 

33.79     °f  2.7.i     °'-' 
34.48     °^'>  ^.(14     °" 

3  1.1     °  '° 

0.09 

0..;.'. 

0.9(1 

0.9,'> 

404 

97.  (!S    °''-' 

^  ,^     o.oS 

3.27     °" 

0.9(1 

0,.;.', 

0.89 

0.9(1 

0.94 

o.os 

0.6y                    0.12 

0. 1  J 

1 

4o:> 

9><.fl2 

2.4.' 

3,'..17                2.5-J 

3.39 

0.93 

0.(!4 

—0.89 

—0.1)7 

4oi; 

'■')  .'■i7+°''^ 

2  31  — ^■°"' 

,•!.->.  87 +°- 7°  2  41-°" 

3.51+°  '^ 

o.no 

0.(14 

0.811 

0  91) 

407 

I.'«V53     "''"^ 

»';>■     —0.02 

;i(;.58     °;'    2  31      °'° 

.l..;4     °'-5 

0.87 

(l.(',."i 

0.81) 

101 

40^ 

1 01,. 50     °''7 

,"'!' ,    0.00 

37.:il»     "  '-    2.20     °  " 

3  78     °'l 

0.«4 

o.r,.'> 

0.90 

1.02 

409 

102.47     °"7 
0.79 

0    .,,,+0.0.! 
0.05 

38.02     °'' 
0.7J 

2.10  ° '° 

o.o<; 

3.93     °'i 
0.15 

fi.8(» 

9.i;i: 

0.90 

1.03 

410 

liiiir, 

-      '  <      .                0 

•'"*"'  .      . 

201 

4.08 

0.77 

o..;7 

—0.90 

-    1  01 

411 

l(.|.45+°'"> 

•>.V2+°°'' 

3-1  47  ^-°  7.?    1  l.;|-°°« 

4  .,.( +0.  (^ 

0  74 

0  (!!. 

0.91 

1.(15 

41-J 

1 ...-,.  41     "'^'^ 

2.i;i    °°'' 

40.19     °'-    IS.;     °°7 

4;  39     °'^' 

0.72 

0  70 

0.91 

1.07 

413 

\w\A\      '°° 

V'.72     °" 

4.1.92     °  '•'   1.79     °°' 
11. .1.1     °  7^   1.73     °°'^ 

l.,5(i     °'7 

.1  .10 

0,7-' 

0.92 

l.OS 

414 

I..7.I      '°° 

•.  s.^    ° '.? 

4.73     °'7 

0.  .1.1 

0.74 

0.92 

1.0!) 

'            1  00 

0.15 

o.7i              0.04 

o.iS 

41.') 

MW  II 

3.1)0 

4  2  39                1  r,9 

4.91  . 

(MI3 

0.77 

— 0.1(2 

—  I    10 

4  Hi 

Hi;il-,  +  '°' 

.,„+o..8 

43  12^°- 7.?  ,  r„-,-°°-«   ■..0!l+°"' 

...111 

0.79 

0.9:1 

1     10 

417 

ihi.t.i     '■°' 

:».38    °-'° 

4:1. SI     °T    l.f,3     °°-'   .5.27     "  "'^ 

0  ..ll 

0.S2 

0.1)1 

1    10 

4H 

111.(7     '•°' 

3.  no    °" 

41,57     °7.<    ,.,.,„     °°-'   .-,.4.1     °  "> 

(1.5.1 

0.  sr, 

0.9.1 

111 

4ri 

"^'''  ;:: 

3«*5     "■'■,    45.29     °7'    1.59     °  °'    5.fi.-,     °  "' 

0.  j;                   0. 7.'              0  o'l               0  I,) 

It, v. 

(1.88 

0.97 

112 

« '.'0 

ll.i.51 

1.12             j    4''.. til                l.,'!»               •'  XI 

0    .'lii 

(1  n2 

--.111-' 

1   1  J 

' 


T  11  K    O  li  15 1  T   OF    U  i;  A  N  U  8. 


361 


TAIJLK  X,  Alio.  'i.—Vontinued.                                                          1 

Arg. 
430 

(i-.c.o)  Diir. 
ii:i.r.:  , 

(0.8.1)    Diff, 

ff                    tt 

4.12  , 

(r.f.l)    Uiff. 
40.01  , 

^«.n.2i  DilT. 
1.59  , 

{K.c.i)  Diff. 

t'                 tt 

5.S4  , 

(.'.X.3) 

(0.C.3: 
0.92 

(u.g.4)  (('.(•.4i 

tt     1        tt 
_0.9S  _1.I2 

0.53 

421 

ll4.r.:i  +  '°-' 

4.42+°-^° 

40.71+°- 7° 

1.00+°°' 

0.01+°-'° 

0.52 

0.05 

0.  99 

1.12 

4 -2  J 

Il;Vr,5     '°' 

4.7;i    °  '■ 

47.41     °J° 

'•'•■■«     m 

0.23     °  "-» 

0.51 

0.09 

1. 01 

III 

4-2.i 

iii;..-,7    '•°-' 

.'■..07     °" 

4S.I0     °-^,'> 

1.05     °-° 

0.42     °"> 

0..50 

1.02 

1.02 

III 

424 

117..V.     ■°-' 

f>.42     °-^l 

4S.7'J     °"^ 

1.09     °-°/ 

0.01     °  "> 

0.40 

1.00 

1.03 

III 

1. 00 

o.;,,S 

0.69 

o.oO 

0.19 

42.'-. 

IIS.-,!) 

."..SO 

40. 4S 

1.75 

O.so 

0.40 

l.IO 

—  1.04 

—  I.ll 

42(! 

ii'.i ,-;!»  + '■°° 

(!.2l+°-»' 

50.15+° '^7 

1    sl+°°'' 

O.OS+o.l.S 

0.49 

III 

1.00 

l.IO 

427 

I2(t..-.!)      '  ■'° 

O.C.'J     °42 

50., SO     °-^5 

l.SS     °°7 

7.17     °"> 

0.40 

LIS 

(OS 

1    10 

42S 

121.00      '•°' 

7.07     °-»> 

51.44     °^'-^ 

1.00     °°" 

7.:i5     o«« 

0.50 

1.22 

1.1 10 

1    1(1 

421> 

122..-)9     °'''' 

7.54     °^7 

52.00     °^'^ 

2.00     °'^ 

1.52     °'7 

0.51 

1.20 

1.10 

1.(19 

O.IJ<) 

0.49 

0.61 

0. 10 

0.1. S 

4:i(» 

\-l:i.M 

s,o;j 

52.07 

2.10 

7  70 

0.53 

l.:!0 

—1.11 

— 1.(19 

4.11 

124.;'i7+°'''' 

K.-,4+°S' 

53  ■>7+o."0 

2.20+°'° 

7.S7+°'7 

0.51 

1.34 

III 

l.OS 

4.'!  2 

1  ■.'.-..  ;.5     ^y'"* 

o.o.s    °5-* 

53;s5     °-.S« 

2.:is    °'-  H  o;i    °"' 

0.50 

I.3S 

1.11 

1  (17 

4:!:i 

I2(l.!ij     °'>7 

it. 02    °  ^» 

54.41     °S6 

2.52     °'-»jS  10     °i^' 

0.5  s 

1. 41 

1    12 

I  (ii; 

434 

127.48     °'>^ 

10.  IS          -:  ' 

54.05     '■•''' 

2.00     °'t    S.31     °'5 

0.01 

1.45 

1.12 

1.0.. 

o.nO 

*'.  -,;■) 

0-5.3 

"•'.i 

0.14 

4;l.i 

12S.44 

to.  71!        \ 

55.4  s 

2.  Si 

s.  IS 

0.01 

1.4S 

— 1  13 

—I. (11 

4. If. 

l2'.».:ts+°  ''t 

,,  .(^4-o.r., 

55.00+°- 5  < 

2.00 +°-'-^ 

s.i!2+0M 

0.00 

1.51 

1.1:1 

1,(12 

4:n 

l.so.ai    °'>.' 

11.00    °^'^ 

.-.o.is    °-»" 

3.13     °'7,s.74     °'-' 

0.70 

1.54 

l.l:i 

I.IKI 

4. 'IS 

l:!1.2l     °''.' 

12.02     °^3 

50.04     °-t'' 

3.:to    °'7  s.so    °'- 

0.73 

1.57 

1.13 

0  09 

4M 

1.12.1  A     °'" 

13.20     °'^'l 

57.:!S     °H 

:i.47    °-'i's.!i7    °" 

0.77 

1.59 

1.13 

O.US 

0.91 

0.67 

0.42 

o.lS 

0  10 

410 

i:!:t.o«  ,     „ 

l.!.o:i 

57. SO 

:i.05  , 

0.(17 

0.S2 

l.ol 

—1.1:1 

— 0.07 

4tl 

l:!;i.i).-.+°-^2 

11.01 +^^''< 

5s.i.)-l-°-.V) 

3.si+°"' 

0.10+°°'' 

o.Mi; 

1.0:1 

1.12 

0.05 

4t2 

l.!».s;i    °'^^ 

1.-..31     °  7° 

5S.50     °-'>7 

1.01     °-° 

,,04      o.oS 

0.00 

1.01 

1.12 

0.9:1 

44;i 

l;i.V70    °-;^7 

10.01      °7° 

5-i.OO     °-5^ 

4.21     °-° 

iiTlh    °°7 

0.05 

1.05 

1.12 

0.92 

444 

i;i.;..M    °'^.^ 

10.73     °7^ 

,V.I.22     °-'- 

4.11     °-" 

0.37      °°'' 

0.90 

1.00 

1.11 

0.01 

0..S4 

0.72 

0.29 

0.21 

0.05 

44:) 

l.!7.:iO  ,     „ 

n.t.'i 

50  51   , 

4.05  , 

0,42 

1.04 

1.07 

—1.11 

—(1.90 

441; 

l:is.21+°'J^ 

H  l!»+o.74 

.V.t.7s+°'^7 

^^,.fo.2. 

0.45+°°.? 

1.00 

1.07 

1. 00 

0.90 

447 

1:10.02    °    ' 

n.03     °"' 

00. 02     °'-^ 

5...7     "W 

0.4s     °°.^ 

1.13 

1.07 

1.07 

0.89 

44.S 

I.IO.sl     °19 

10. OS     °  "5 

00  22     °-^° 

5.2s     ^  -' 

.,  ,,1+0.01 

lis 

1.00 

1.00 

O.SS 

44'J 

140.,V.»     °1^ 

0.77 

20.4.-.     °77 
0.77 

00. 10     °'« 
°'5 

5.49     '    -''    0.40      '°° 

0.221              —0.01 

1.23 

1.05 

1.05 

0.S7 

4.''>() 

141..1C.  , 

21  22  , 

00.55  , 

5.71   ,           M).4S 

1.2s 

1.04 

—  1.01 

O.SO 

4.-.  I 

l»2.li-^°7.^ 

21.00+'=  77 

00.07+°-'' 

5.02+°-"    ;t.«5-°°.' 

1.32 

1.02 

1.02 

o.so 

«r>2 

11 2.  S.J     °;- 

22.77  '-■'■;; 

(;o.70    °°'^ 

0.13     "■"    0.42     °°' 

1.37 

1.00 

1.00 

0.80 

l.-,.l 

ii.i.:,4   °y 

23.55     °"'^ 

(;o.s2    °°'^ 

0.34    °'"   o.;is    °''4 

1.41 

1.5s 

0.!)0 

O.SO 

.:>4 

144.2;j    °-^'' 
0.6.S 

•^^■-■^  :S^ 

0O.S5+°°-' 

0.00 

0..55     °-".0.:!2     '^•=^' 

0.20|                  0.07 

1.45 

l.f.5 

0. 9S 

0.80 

4."..". 

lU.'.M  ,      , 

2.-..  11  , 

00  s.-> 

0.75  ,            0.25 

1.49 

1.52 

—0.97 

0.8(i 

4.">tl 

ii.-..r..i+°^'5 

2.-..so+°'« 

oo..s2-°-°;1 

,,,,,.+0.2.    ,,,7_o.oS 

I.. 53 

1.49 

0.95 

0.S7 

4->7 

■4 '. ; 

110.10    °^'' 

20.07  -:■ 

27.45     "1; 

0(1.70     °°" 

0    10 

7.15     °-';'    0.07     °'° 
7,:!5    °--S:s.07    °-'° 

1.57 
I.OU 

1.45 
1.41 

0. 9;i 
0  91 

(I.S7 

(i.SS 

-i^V' 

147.40     °-" 
0.5a 

■"     "77 

"^-»     o.,6.-"'     o.,7 

S.SO     °-" 

0.12 

i.oa 

1.37 

0.90 

O.SS 

4(11 
4ii2 

4<;:! 

4(;i 

147. OS  ,        , 
ils.r,4+°-S^' 
110.07     °-5^ 

1.0 .V.I   °;- 

i;,o.os    ^••♦'' 
0.46 

2S  Oil  , 
20.75+°"^ 

30.51  y!: 

31.20     °i5 

32.00  °;] 

00  :is          „  '7. '70  . 
0...20-°-'''   7.S7+°-'7 
,V.(.OS     °"   s..      °-'/ 

^y.4o     o.jii'^-'-'     0.,- 

S.7I 

s.(io-°'-» 

S.40     °-'-» 

S31     °'5 

S.14     °-'7 
0.17 

i.r.R 
1.0s 
1.70 
1.72 
1.73 

1:13 
1.29 
1   24 
l.IO 
l.lt 

—0.89 
o.ss 

0.87 
(ISO 
0.S5 

— 0.S9 
(I  90 
0  91 
0.92 
0.93 

4f..-. 
4  liii 

l."iO..->4  , 

l.-.o.os+°-'-» 

S:l^°'^^ 

50.  U     „,Jx.40,„,, 
.5s.sl-°--"   s.02+°-'-' 

7.!'7          ' 

..,,_o.,,S 

1.74 
1.75 

1.09 
1.0:1 

—0.85 
0.85 

—0.94 

0  00 

4i;7 

4  (is 

4i;'.t 

'■'•<"     Mo 
'•■"•-'     OS 

•^"^  ■'•'  0.C6 

,5s.41     °-" 

::::  °-" 

7.01     °'^ 

7.12     °"' 

7.22     °-° 
0. 21 

1.75 
1.75 
1.75 

O.OS 

0.9:1 
0.S7 

o.sr. 

0.S5 
0.85 

(l,!IS 
1   (10 
1  (12 

470 
471 
472 
47.) 
«74 

Si  3 

^^     °:^  B  ^^ 

•••'•"'••     o.5,S   "•-''     0.03, 

7.01 

O.sl   -°'° 
0.00     °-' 
o.:is    °-- 

,,    ,,,      0.22  p 
"■'"      0.23 

1.73 
1.72 

1.70  : 

I.  OS 
l.Olt 

0.S2 
0.77 
0.72 
0.07 
0.U2 

— 0.S5 
O.SO 
0.80 
0.87 
0.87 

—  1  (13 
1  (15 
1   ((7 
1.(10 
I    1(1 

4',.') 

l.->:i.!i'i 

3'».24  , 

51.51        .     0  :;o  ,         ' 

5.03 

i.r.3 

0..'iS 

—O.SS 

—  1    II 

47; 

l,-.».l(!+°-'' 

30.70 -t-°-5.S 

.,.,., j_o.r.o  ,,.,,+0.0,   -  .,-0.22 

1.00 

0.54   1 

0.00 

1   12 

47". 

l.-.:i.:il     °''^ 

10.:..'    °-5.5 

5:1  -'s     °''.'   <);i"+°°'    -.  4S     °-^' 

1.50 

0.50 

0.91 

(    1:1 

47'* 

ir.i.-.i     "'7 

10.  s..     °-5o 

,-,.,  ,;.,     °<><>  ,)  .([-o.oi    .^  .„^     0.2.} 

1.53 

0.40 

0  92 

1   1  t 

^.It 

i:.i..;o    '■  '.^ 

0.46 

5i;i»  "''-^  oN.'o  °°-'  5;(';3  °»- 

1.4s 

0.42 

0.9:1 

11'. 

n.  1  1 

0.(1. 1                              O.O-     ;                          0.  2  1 

1 

4. SO 

l.M  77 

11  77 

51  25               0.25              '4.S2 

lit 

0.;!9   \ 

-0,94 

-l.IO 

262 

Tin:   oil  HIT 

OF    V\\, 

i  N  U  S. 

T.U'.I.K   X.  -Villi,  W.—Viiiihininl. 

U',r,4 

Arg. 

(u.c-.o)  uiir.  i 

{V.g.l)      l>iff. 

ft              If 

(u.f.i)  i»iir. 

[f.»<.2)  l>iff.(r.<.2)l)iff.| 

(I'.H.a)  (o.f,3) 

((•.-.4) 

„     - 

H                   H 

„-         „     1 

„ 

^'^O  '1-.4.77  ,              41.77.           1 

M.^."!               9.2'.             1 

4.S2 

1.44 

0.39 

0.94 

—  1.1.1 

^'^l     l.-.4.«l!+°°" 

42.20 +°« 

;.o.r.3-°-7-'  9.20-°"^ 

4.C.0-°'" 

1.39 

0.311 

0.911 

l.lll 

^-^^    i:,4,y:j    ''•°' 

42.111     °-»' 

4!l.S0     °-7.!'!l.l5     °°-^ 

,,,,       O..M 

1.;;:. 

0. 3 1 

0.98 

1.1.1 

<^''>     l,V».t)s    °°5 

4..  .,,,    °  .}» 

49.04     °'^!   9.0s     °-^7    ,  ,,     0.22 

1.30 

0.31 

1.00 

l.lll 

<»*     154.«'.l+°°' 

4'!'.:i4     "''-^ 

4S.,.8     "•J"   9.00     °'°'^   3.9(1     °'' 

1.2.-. 

(1.30 

I.Ol 

l.lll 

1              —0.0 1 

■^•i.} 

o.-!<)              o.iOj             0.21 

••'*•''     ir)4.!)S 

•»:'''''1_L     ,, 

4T.19         „      8.90         ,    '3.7.-. 

1.20 

0.2s 

-1.02 

—  I.Hl 

^«''     l.-.4.'J.^-°°-' 

4:t.9H+°-'' 

4ii..;9-°-^°  H.so-°''° 

3..'-,4-°'" 

1  15 

o.:;7 

1.04 

II.'. 

*'^"     l,-.4.'..0     °";V 

4t.--.-.    °:',    ^yM   °«'is..;'.i    °" 

3.34     °=° 

1  10 

0.211 

1.011 

1 . 1 .'. 

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1.20  1 
1.28 

I.:i0 
i.aa 

1.36 

i.a7 

l.;is 

1.40 
1.41 
1.41 

1.42 
1.42 
1.42 
1.41 
1.41 

1.41 
1  40 

i.ao 
l.:is 
i.ao 

i.a4 
I.  a  2 
I.ao 
1.2s 
1.20 

1.24 
1.22 
1 .  20 
1.17 
!.I5 

1.12 

1.09 
1.07 
1.01 

1.02 
l.Oi' 

;..iis 

0.05 

o.o:t 
0.01 

O.SO 
0.N7 
0  s5 
o.s:i 
0.S2 


" 

n 

1.43 

0.80 

1.41 

0.01 

I.a8 

0.0:1 

1.3fi 

0.05 

I.aa 

0.90 

'  I.ao 

—0,07 

1.20 

0.00 

1.2a 

1.02 

'..20 

1.01 

1.17 

1.00 

lit 

_1.0^ 

1.11 

1.10 

1.08 

1.12 

1.01 

1.1;! 

l.Ol 

1.14 

0.08 

-1.15 

0.05 

1.15 

0.02 

1. 10 

0.00 

1.17 

0.S7 

1.18 

0.S5 

—  1.10 

O.sa 

1.10 

O.sl 

1.10 

0.70 

1.10 

0.77 

1.10 

0.70 

—  1.10 

-1.15 
1  15 
I  10 
1.10 
1.17 

LIS 
lis 
lis 
lis 
1.18 

-l.is 
117 
III', 
115 
114 

-1  1:! 
1.11 
lull 

1  07 

1. 05 

.  1  o:i 
1  111 
0.00 

0.07 
0.05 


u.  1  i 
0.72 
0.72 


1.17 

in; 

1.15 

111 


0.71 

— i.ia  . 

0.71 

Ill 

0  71 

1  110 

0.71 

1.07 

0.71 

1.05 1 

O.sl 

O.SO 

0.70 

0.7s 

0.77 

0.71! 

0.70 

0  70 

II 
0.70 


070 

II  77 

0.7s 

0  "|S 

0.70 

0.80 

0.71 
0. 7  2 
0,72 
0.7:1 
0.7» 

0.70 
0.77 
0.7s 
O.sii 
0.s2 

0.S4 
O.sr, 

0.S7 

0.00 
0.92 

0,04 
11,00 
II.  OS 
1  nil 
1.02 

1.01 
1  00 
I  IIS 
1    10 

1  l:t 
lift 


-1.01 

1.02 
1. 10 

0.0  s 
0.00 

-0.01  . 

11.02 
0.00 
OSS 

0..S7 

-O.sO 
0  si; 
0.S5 
O.SI ; 

0.S3 

o,s2 
0,s2 
0  S2 
0  s| 
0  s| 

-O.sl 
0.S2 
ii,s:i 
o.«l 

0.s5j 

-0  s5 


0.9a 

0.01 

0  so 

0.S7 

0.80 

0,S5 

0.S5 

0,s| 

O.sa 

U.S2 

0.81 

o.sl 

o.sl 

O.SO 

O.SO 

0.80 

o.sl 

0.82 

o.sa 

0.84 

0.85 

0.S7 

0  SO 

0.01 

0.U3 

0  94 

0  00 

0  OS 

1.00 

1.02 

1  04 

1  00 

l.OS 

1    10 

1  12 

-i.ia 

I 

■ 


264 


TUB   ORBIT  OF   URANUS. 


Arg. 

rAIU.K  X,  A 

«).  3.  —  t 

oiilinuvd, 
(p.c.O) 
710 

(p-c.O) 

r.L'4 

(p.«.l) 

(p.C.l] 

80 

(p.«.8) 
40 

(p.r.2 

Arg. 

(f.B.l){,.C.l] 

17     \     244 

{p.*.8)|(,.r.8) 

0 

I  MO 

5S 

no 

■33     !     10 

1 

y>:, 

I7S 

!)0 

.^.0 

.■-..s 

nl 

747 

17 

240 

32          II 

•2 

:>-2:, 

171! 

!)0 

.50 

.'.T 

n2 

74S 

10 

1     Of,;, 

30     ,     1 1 

;t 

r.L>i; 

174 

111 

.')! 

C>1 

03 

740 

10 

!    aoo 

20          1 1 

4 

:>-ii 

17:J 

ill 

1 

M 

04 

740 

10 

iiOO 

27          11 

:. 

y>H 

170 

J)2 

.--..•l 

.i« 

0.5 

:   7.50 

10 

:    271 

80         13 

<; 

:.■-'!» 

ins 

n 

.'.4 

f..-. 

no 

7A0 

ill 

277 

24     1     13 

7 

r.:il 

ni'i 

W'i 

,S.') 

.'ia 

in 

7.50 

17 

2«.2 

23          13 

n 

;>:i:i 

iii;t 

nu 

:,:, 

t,i 

OS 

7.50 

17 

2S7 

22     1     14 

!l 

6;i.j 

nil 

93 

.'■ill 

C,:\ 

CO 

740 

IS 

203 

20          14 

III 

5:ls 

l.^)S 

!)| 

r.7 

r.:i 

70 

7is 

10 

31IS 

19          15 

II 

.Mo 

IMl 

!)4 

r>7 

.■>J 

71 

717 

21 

803 

IH          10 

12 

:.4;t 

I.VI 

O.-i 

.'.s 

.M 

72 

7111 

22 

."JOS 

17          17 

i:i 

.')4C, 

i.".i 

on 

.'■.!) 

.Ml 

7:i 

7U 

21 

813 

10          IS 

u 

&.')0 

11!) 

Art 

t>[) 

4!l 

71 

7*2 

21! 

iUH 

14          19 

\:> 

r.:.;! 

1  Hi 

07 

f,n 

4S 

7-'> 

710 

20 

323 

!3           20 

Hi 

:,:r, 

III  ! 

!)S 

no 

47 

711 

737 

31 

327 

12           21 

IT 

fii;ii 

Ill 

!M) 

ni 

411 

77 

734 

34 

3;!  2 

1 1           2.2 

IS 

.Mil 

|:is     1 

llll) 

ni 

4."> 

7'>' 

731 

37 

3311 

10        2:1 

ll» 

fjI'.S 

i;ia    i 

lol 

v,-2 

41 

711 

72S 

40 

340 

00      1     24 

:.'it 

r,7:i 

|:!-J     1 

102 

r>i 

44 

SO 

724 

43 

34  4 

00     [     20 

ill 

.'•77 

1  J'.l     ' 

III.) 

n--> 

4-.> 

si 

721 

47 

3  IS 

OS          27 

2-i 

.'.MJ 

1  -Jli      i 

III.') 

n-i 

41 

S2 

717 

.51 

3".2 

07          2S 

'.»;» 

li'^ii 

1  :.>il     ! 

IDA 

nil 

40 

h:; 

713 

.5.5 

3.55 

07     j     30 

'H 

.Ml! 

1 M     1 

IU8 

)i.'i 

:io 

SI 

7iis 

.50 

359 

00     1     31 

■::> 

U'.\r, 

117 

ton 

fi.i 

ns 

s.'. 

7ii3 

03 

3n2 

00     '     33 

M 

i;iil 

ll:i 

in 

(111 

.■17 

Kii 

noM 

ns 

;io4 

0.5     ,     34 

1!7 

nun 

ilii 

II:) 

lilt 

.•!'! 

y-1 

oo;i 

72 

307 

0.5      {     30 

•2H 

i;il 

Iii7 

II '» 

nil    j 

:il 

MS 

nss 

77 

300 

0.5     1     37 

L".» 

r,iT  1 

lii;i    1 

in 

n:i    1 

;!;i 

S'.l 

ns-j 

H-J 

372 

0,5          30 

:io 

C-2-i  ' 

|M0       ' 

in 

nj 

■'.> 

IHI 

070 

S7 

373 

05          41 

;ii 

\■,•J^ 

nil 

i.'i 

ii'j 

:n 

111 

070 

02 

a7i'> 

0.5     1     12 

•M 

n.-ia 

!i;i 

\i.i 

•1:! 

:iii 

02 

on3 

07 

370 

U5     1     4 « 

.'t;i 

li.'tM    ' 

1)0 

l-jn 

(\-2 

'J!l 

oil 

0.57 

102 

«7s 

05     '     4.. 

:ii 

r,4,l 

Xli 

1^'!) 

t;\ 

'js 

01 

0.50 

Ins 

37S    ! 

00          47 

li.) 

i;4x 

S'J 

n-j 

n! 

•J  7 

0.-, 

04.! 

III 

370  ; 

00          4s 

ait 

fi;)4 

7!) 

m.-i 

I'll 

•jii 

on 

o:!it 

110 

370 

on    '    .50 

:i7 

c.r.ii 

7 11 

l:i!) 

I'll 

•Ji 

'n 

020 

12.5 

379   i 

07      i     .51 

;is 

11114   i 

7J 

ll-.> 

.^!l 

L>:i 

iH 

022 

130 

370  ' 

OS           hi 

;;:) 

Clio  j 

nil 

iir. 

f)*! 

:.'2 

O'.l 

014 

|:in 

370 

OS           ,5t 

III 

1171 

i;.') 

Ill) 

.'iS      i 

21 

tnil 

mm 

111 

Ul'* 

II '.I         il.'l 

II 

«><(» 

l'-2 

i:.j 

.'.7 

211 

I'll 

.5'.H 

117 

.377 

III           .'>ll 

ij 

r.>..'i 

:<H 

1  Ml 

.'ill 

1' 

1 112 

.500 

153 

370   ! 

1 1       .57 

i:; 

t;-i'.i 

!,:, 

li;i) 

f  * 

;  . 

lii:i 

.5.  J 

I..0 

.174  ! 

I J            f.s 

1 1 

ii'.il 

.■>:! 

h'.l 

.M 

M 

in, 

.57  1 

in4 

«73 

|:i         1:0 

•i:. 

iI!W 

4'i 

1  (IS 

.'.:l 

17 

III.'. 

(in  .5 

170 

871 

14         ni    . 

ii; 

Tii:; 

in 

171 

:<i 

]'■■ 

Hill 

f.:.7 

170 

3ns 

15         n-j 

17 

T'lT 

1.! 

177 

..1 

i:. 

107 

.51. 

IM 

300 

in        02 

IN 

711 

Ii> 

!'-J 

."■II 

II 

Ins 

^..r» 

ixn 

3n:)  ' 

1 7           03 

Vt 

71.'. 

.■17 

Isil 

f< 

1 1 

||  || 

.5.10 

102 

.300 

1 0        n  1 

:,ii 

:i'.i 

:i.". 

lill 

4  7 

1 

III! 

r.2i 

107 

3.57   1 

20        n  I 

:ii 

7  ■-'•-' 

;ii' 

1  nil 

■('■• 

l:; 

Ill 

M2 

20-i 

3.5;i 

21          05 

5  J 

7  X 

;;i) 

•.'111 

I  1 

IJ 

112 

.5(i:! 

207 

.'(.•.I 

23          115 

.■i:i 

7i!;i 

".'■' 

■Jim; 

41 

IJ 

111 

401 

■J  1  J 

:un 

24        no 

M 

7-IJ 

-'ii 

::!1 

IJ 

1 1 

Ml 

4^.1 

217 

:a2 

2;.        no 

•i.'i 

7;i."> 

•Ji 

•Jill 

|i» 

11 

ll'i 

47.". 

222 

33S 

27         ii« 

Mi 

t  ■'  I 

•ii 

'Jlil' 

:i;i 

1 1 

ll.i       i 

4n.5 

227 

333 

2s    1    07 

.'.T 

"ilo 

•>\ 

ir, 

MS 

11 

117 

4.50 

231 

320 

30     <     (17 

.'i>< 

7  1  - 

I!) 

•i-\:\ 

:ir, 

11 

IIS 

itn 

23.5 

324 

31          n7 

:iii 

711 

IM      1 

a.!s 

:!.'• 

I'l 

ll:i 

4:in 

230 

310 

32          07 

r.ii 

:ir, 

17    ! 

1 

.'1 1 

10 

Ijii 

'-•• 

Ji;i 

.ill 

:;l 

! 

tin;   OIUJIT   OF    UUANU8. 


965 


A.K'.     i  (p...'.0)'  (,.».l)\  (p.o.l)j  (h.».3)  ((...'.a) 


TAIiLK  X,  Ami.  :t  —Couiinutil 

Arg.     ,(p.r.Ojj    (,...1.1)     (,..i-.l)(,..«.2)  (,..., L') 


!20 
1^1 
IJiJ 

1 2.1 
i:2i 

1  i:) 
lii, 
lil 
IJS 
l^-.l 

l;io 
l.'tl 

\M 
V.VA 
1.14 

I.!.-) 

(•■IC 

i.n 

l:is 

no 

I  ti 

in 

1 1.". 
!  n; 

M7 
IIS 
ll'.l 

i.-.o 

l.'.l 
l.>i 

I ;.:( 
I.M 

I.Vt 

I.m; 

l.-.i 

i:.s 

iii'l 

i.;a 
i<3 
lr,| 

1.;.-. 

I...; 
u; 

|i;s 
!  ll'.l 

'7i> 
171 
IV2 
17:1 
171 

17  . 

i;ti 

17X 

i:'.« 

lot* 


I:f7 

u; 

{(17 

.•(!t7 
:H7 

:17s 
.I'l.s 

a:.'.) 
;it!» 
;i;i!t 

.lliO 
;i;;!0 

:no 
:tot 
•-'ij 

■isj 
27:1 

•-'.".I 

L>.li> 
227 
21s 
2(r.i 
.Ml 

I  ;i:: 
l-t 
i;i; 

his 

h;u 

I. .2 
i  I.". 
I:i7 
!.,() 

in; 
I  (1:1 
Ilia 

!Ml 
-•4 

71 

7:J 

•17 

i;d 


;iii 
:,i 
2'.» 
2)1 
2:1 

}\ 

IM 

!•: 
u 
u 

I! 

Auk, 


8  4:1 
241 
:if>U 
iir)4 
SI57 

'2110 
2*'>:i 
■ilir. 
iiISM 
270 

27  a 
274 
27:1 

277 

2i.S 

27!> 
27!) 
2Kil 
'.'SO 
J^ii 

2S(I 

2S(i 
2-|!l 

27> 

277   '■ 

•!'■< 

271 
.)-■> 


2''.!' 

2.;i 
2';fi 
2i;;! 

2r,(i 


<1'.i 

a  If 

21a 

21" 
•Xl'i 
*•!.< 
i2'.> 


:.: 

22''> 

:,\\ 

2:i2 

4v 

JI'i 

1  t 

21  . 

«:) 

212 

2«s 
2111 

211I 

i!i; 

I'.il 

1 .1(1 
ls« 

IfJ 

I7'.t 
17:. 

I7i 

,  llt7J. 


814 

30M 

2!)H 
■in 

2»«<1 
2«0 
274 
2C>S 
2*1-2 

2.)P. 
2.)(l 
244 

2:!S 
2:12 

2 1  It 
21.; 

2117 

•.;oi 
I'.iri 

ISO 

isa 

!77 
171 

n;.-. 

Ii-.O 
I.M 
14!) 
114 

i:!i) 
l:i4 
1 2!l 
I  J I 

!  m 

ii.\ 
111 

1(17 
10:1 
1(10 

'.I'l 
i».i 

!)() 
>'7 
••» 

H| 

ill 
77 

7.'. 


Kli 
li.S 

lilt 


34 

a:> 

37 
AX 

a» 

41 
42 

4:1 

44 

4.1 

A(, 

47 
4S 
4.) 

r.o 

.'.() 
:.i 


w.\ 
f.a 
f.a 
.'•a 
fia 

ria 
f.a 

f.2 


I'll  I 
4I> 
4'.( 

4S 
47 

1 1. 
4.1 
4.'. 

41 

4.1 
42 
41 
40 

as 

3<S 

a.i 

a  I 

aa 
1:1 

a  I 
ai) 

2*.» 
2«< 

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at 

t* 

w 


fir, 

ci; 
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05 

(14 

'■.;'/ 
ca 
i;2 

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,'iS 
.'>7 
;■>(> 


.M 
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4!l 

47 
41; 
4.". 
41 

4." 
41 
4" 

■IS 

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a.'. 

.•!( 

;;._> 

;;i 
;;i» 
,;ii 
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^'!' 
2s 


27 
27 


2s 


ail 
•il 


1S(I 
IHI 

iba 

IIH4 

Is.-i 

I. si; 
l.s7 
iss 
IxH 

i!MI 
11' I 
!!>2 
l!ia 
l!»4 

m.-, 

I  lie 
I'.t7 
1:1s 

r.c.t 

2('0 
2(1 1 
2112 
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2111 

211.') 
2iir. 
2(i7 
2(IH 
2(1!) 

210 
211 
2 1 2 
21:; 
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2 1  .■> 
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2  IS 
219 

22il 
221 
2"*2 

2. 'a 

22  4 


221'. 
227 
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!.!•.".» 

van 

2!  I 

2a2 
aaa 
2a  I 

2:1.^ 
yaii 

217 

2as 
2:i;i 

24(1 


11 
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14 

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in 
II 
1:1 
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21 
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2(1 

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ac 

a:> 
4.1 

17 


7". 

Sll 

K.'i 
HI 

!H'. 

I(i2 

IMS 

ill 

12(1 
1 2'; 

laa 

1  ••.-.» 
11.'. 

1.-.2 
I.'.!' 

Kill 
17a 
ISO 
l>7 
!•'» 

Jill 
2(is 

2  HI 
22a 


24.. 

2.'.a 

2(10 

2  (IS 

27.-. 


.M 

(10 

1    i;.'! 

(1!) 

172 
Ids 
1(1.^ 

Kia 
ir>i> 

l.^r, 
I, .a 
i.-,o 
H7 
III 

Ml 

las 
1  an 
laa 
la  I 

12!) 
127 
I  2.". 

12a 

121 

120 
IH 

Hi 

iii; 
111 

11:1 
112 

112 
III 
110 

ltd 
11(1 
110 
III) 
110 

11(1 
I  III 
I  111 
111 
112 

lia 
III 
II.. 

I  HI 
117 

US 
II -.1 

120 
122 
12a 

12.-. 

12: 
12^ 

ia;t 
1. 1 2 

lai 
I  an 
lax 

1411 
142 

I4« 


(1(1 
fiO 

117 
(17 

(l.S 

no 
71 
72 
74 


I  — 
I  " 
I     W 

'      Ml 
N4 

m 
so 

!i| 

04 
!t7 
!I0 

102 

III.'-. 
lOH 

112 
II.'i 

IIS 


I  a.'. 
120 

l.'!2 

lao 
14a 

He 
l.'.O 

I -.a 

i:i7 

ll!0 
Ml 

Ids 
171 

17:. 

17s 

isa 

l«»s 
102 

l!i:i 
los 

201 
201 

207 

210 

21  a 

21  d 
24  » 

ast 

244 

22d 


2a  I 


2(. 
25 
25 
24 
24 

24 
24 

2a 
2a 

2.1 

2a 

24 

24 
21 
24 

2.'. 
2."> 
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27 

2s 
2s 
20 

ao 
at 

32 

:ia 

34 

a.', 
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as 
ao 

40 
41 

42 

41 
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41'. 

47 

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40 
.')() 
60 

U\ 

Ui 
WA 
.'.I 
.'.4 

.'.5 

f.r. 

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.^r, 
r.7 

r.7 

f.7 
.'.s 
.'.s 
fix 


33 

aa 

34 

;!.•. 

3U 
37 

as 

an 

40 
41 

42 
43 
44 

45 
411 

47 

4S 

4!( 

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u\ 

Ui 
52 
53 
54 
64 

55 
55 
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57 

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57 
57 
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57 
57 
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57 
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M 
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55 
55 

54 
53 

52 

52 
51 

51 

5«» 

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4!l 

IS 

4: 

ti. 
I.. 
I.I 
It 

43 


266 


T  U  E   O  K  HI  T  OF   U  11 A  N  U  8. 


TABI.K  X,  Aiiu.  a.—CuHtinuid. 


ArjT.         (,.  <•.(•)  j  (p.«.lj|  (p.c.I )  if.x.'l)  ((..<'.2) 


\  i  'f 


>.>I0 
■Jll 
•J  I J 

2i;{ 

214 

'->  (.■> 

'.M7 

21'.) 

2.>0 
2.-.  I 
2.-.2 

2:>.t 

2.-.t 

2.".  5 
2.'.i', 
2.-.7 
2,>S 
2J9 

2i;o 

21. 1 
2il2 
2i;:! 
2il4 


2i;h 
2i;'.i 

270 


2-iit 
2sl 
2i2 
2H:t 
2-<l 


2s.; 


2HH 

2s'.» 


2',i:i 
2'Jl 

2:t.) 


2;i;i 


:tiio 


27'. 
2s;i 
2:mi 
2 '.IS 

;i2(i 

H2S 

;t.Ui 
ai-i 

;t,")() 
a:.  7 
.'ii;  t 
:i7l 
:t7s 

.IS,') 

;i!i2 
a'.i'.t 
4(h; 
•112 

4!:i 

4Jii 
CiJ 
4:H 
411 

4.'.n 
4.M1 

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I.I  J  11.0:;  0  Jl 

0  07  0.  l.'i  0.  Is 

0    10  Ills  0.111 

0  i.'i  0.1;!  o.J.j 

0,1  J  0.0:1   0.11 

Uii 

<*:>■*  o.ir, 

1.1!  0.0:!  0  jr, 

0.07  0.  Ill  0.17 

0.  Ill  II.  |.S  0.  Ill 

0  JJ  0.11  o.;!i 

0  1  1    0.011     0.10 

40 

ii..'>i  0.17 

1. Ill  0,0:!  O.JS 

o.OH  o.k;  o.in 

o.lo  0.1. H  0.111 

o.:!(i  o.lo  (i.;i7 

0.11   0.011    o.uo 

Tio 

o.r.o  o.iii 

l.os  0  01  0  :iO 

0.0s  0.17  on; 

o.lo  0.1  X  0   111 

0:111  (i.lii  11.41 

o.lo  lids     O.OH 

ni 

0.47  0  Jl 

1. 01;,  001  0:11 

0  lis    0.17    O.l.'i 

0.  Ill  0.  |s  0.1 11 

0  .',0  0. 1  Ij  0  :'ij 

0  (111  O.OH   0.07 

711    n.(:;((.-.'l 

I.O([o.o.-,  o,:;j 

0.0:1  0.  |H  0  1 1 

0  n:i  0. 17  0.  H 

0 Cl    0    1:1   (1  ."ill 

0.011  U.OS    0.(1(1 

Ml)    ((.;;',i  o.-jii 

1.00  o.oi:  ii.::i 

0.0:1  0.  |s  II.  1:1 

0  11:1  0.17  0.  H 

0.71  0.1.'.  0 .r.7 

II  IIS  0,(17    II. o;! 

00  io.:j."i  u.i'y 

1                1 

0.'.i7  0.07  o.;i5 

0.0:1  o.ls  0.1  J 

0.011  U.  li'i  0.17 

ii.s7|0. 17|0.7.'i 

•Ids  0.07    0.04 

loo    o.;n  0  :ti 

o.n:i  oiw  o.:tii 

0.0!l'o.lS   (I.IJ 

o.in  o.li;  0.17 

l.Oiid.Jl  II  s:! 

0  117  0.07  1  o;! 

III)    II. -JT  II  :it 

o.hm'(i.o:p  ii.:i7 

o.o'.i  0. I'.l  0. 1 1 

iioi  0. i.'i  0. in 

1.1 1  O.'J.',  o.nl 

•  I  on  (1.(17    (1  02 

liii    ii.-j.i 

0  ;t7 

O.si  oil  ii.:!7 

0.0:1  0. Ill  0. 10 

0,07  01 1  0.  i.'i 

1.  js  0.  Jii  n.iis 

0  01;  o.or,   0.02 

l:to    ii.jii 

0.10 

0.7!l  0.  IJ  0..H 

0  0:1  0. Is  0  no 

0,07  0.1 1  oil 

1.4J  o,:!l  l.iiii 

0  01;  o.on   o.oi 

Uo    0.17  net 

0.71  0.11  o.:ji 

o.o:t  0. Is  0.0s 

o.iii;  0.1:1  0.1:1 

l..'>r,  O.III  1.1:1 

0  o.'i  o.on   o.iil 

irm  ;0. II  0.41; 

com' on;  0  :is 

O.O'I  0.  |s  0.07 

ooi;  0,1  J  0,1  J 

1.70  (t.4.'>  llii 

0  11",  o.on   o.oi 

li;o   11.11  O.V.I 

o.r,:;  0.  |x  ii.:m 

0.11:1   0.  1  s    0  Oil 

o.n.i  nil  0.1 1 

l.s:!  o.'ij  l.j.'i 

II  (II  0  on   0.00 

170    (I.o'.i  II  .'■! 

0..'.7  0.111  0  :!7 

0.0s  0.  17  II  n:. 

o.o.'i  0.  In  II.  10 

l.m;  o..'.>*  1  :!l 

0  III  o.iin   0.00 

Ho    0.0;  0., 14 

0.,'iJ  o.JI  o.:i7 

o.iis  0. 17  no) 

ii.ni  0.1(1  1  .0:1 

J. oil; o.ii.'i  l.:iii 

0,01  (l,Oll    o.ou 

I'.io    o.ii.'i  o.'ii; 

1 

0.4r.  o.J:;  o.:i(; 

o.OM  0. in  0.114 

U.oi  0.011  'l.os 

J.  JO  0.7  J  1.10 

0,01  0.07   0  00 

2110   'o.lll  ll.fiil 

0.41  o.'J.'i  0  :!') 

oo-i  o.li;  not 

0,0:;  0  OS  0  117 

J.3|!o.7n  1   i:i 

II. 0  1  0.07   0.00 

2I0  |o.(i:i|o.r,| 

0.:ii;  0  ji;  o.:;i 

II  07  o.I.'i  0  IIJ 

0.0:!  0  07  Ollll 

J.  40  II.  n.'i  llii 

0  0|   0,07     (MIO 

2 JO  ;o.iij  or,;! 

0.:iJ  0  J'-  0  :',J 

0.07  0  1  1  oil:.' 

II, IIJ  oil''  0,0  1 

J.4x  0.11  J  1    IS 

0  n|  0,07    II  111 

2:!0    11.11  J  0.1,1 

O.j;  0  :!ii  0  :;i 

0  07  11  1:;  ii.oj 

O.OJ  0  0  :  O.O'i 

J. '■,:",  O.lis   1  .',0 

0  01    •!   IIS     (Idl 

210 

O.oJ  o.ci; 

O.J:;  0:11  0  :;o 

0.01;  0.1  J  o.oi 

II.OJ  O.U.I  O.OI 

J.GI.l.O.i   l..",0 

OdI  O.OM    0.02 

2r>o 

ii.oit  0.1,7 

O.JO  o.:ij  o.JI 

0.01;  0.1 1   II. Ill 

0.01  n  01  0.0:1 

2.f,.'i  I.IO  1  .Ml 

oil.'")  0.0s    0.02 

2i;o   o.ot  0.1,7 

0.  |i:  0:11  0  Ji; 

o.oi:  oil  II  III 

0.01  11.01  0.0:1 

J.  (is   1.1(1   1    IS 

0  0'-  (Mill     O.O.'I 

270  'o.oi;  o.{;s 

0.1 1  0.:;.,  o.JI 

on.',  0  10  II  111 

0  III  00:;  0  (ij 

J. 70  1.21    1    (d 

Ollll  odii    0114 

2^0  lo.o."<  o.r.s 

0. IJ  o.:i.',  0  jj 

II. II.',  O.O'.I  0  (l| 

0.0(1  11.0:1  ooj 

J. 70  1.2.,  I,i;i 

0.0(1  0  (111   o.o.'i 

2!I0    O.lli 

o.i;.s 

O.lo,o.;ir,|ii  Jl 

O.OI   0.0s  0  111 

0.00  O.OJ  o.oi 

J.Gs  l.Jli|  1.  10 

o.on  o.lo   II  llii 

:<oo    0.1:1 

(I  r,7 

0  o«  0  .n  0  m 

0  01  0  07  0  (IJ 

0  0(1  0  nJ  0  o| 

J.c.'i'i  ;!:;  I.:ii; 

•  1,117  0  In    11  (17 

:iio    o.ic 

o.i;7 

{)AU  ii.;i7  ".  n 

0  (1  1  O.OIl  II  0.' 

0.00    O.OJ   0   III 

J. Ill   l.;i.',  I.:il 

•  1  lis  (1  In    11  IIS 

;i'.>ii    0. 1'.i  iMiH 

O.Os  0  ;!;  0  in 

0  (i:!  n  (i.'i  0  oj 

11,00  n  nJ  n  111 

J.. ■"..■,  i.:!7  l.J-"> 

II  lis  oil    0  nil 

:i:io  io.-jj  o.r.  1 

O.O'.i  0  :;7  II  1 1 

o.ii:;  0  (Il  no:: 

0  on  onj  0  III 

J.4.S  i.:!ii  i.r.i 

n  on  01 1    0  III 

;iio    o.ji;  o.i:;i 

O.lo  0  :i7  0  IJ 

nil.'  n.ni  n.ni 

0,00  o,(iJ  0  (II 

J.  40  1.40  1.1  :i 

0.0:1  O.II     0,11 

S.'So 

0.:!IMM',1 

0.1  J  11  :!i;  0  10 

o.nj  0  11:!  0  (i| 

(inn  0  OJ  0  111 

J..11    1    40    1   0(1 

Old  0  IJ    0  1 J 

;ii;ii    0.:!:;  0  .','.1 

0.11  II  :n;  II  o;i 

o.oi    00:!   0  (I.'i 

0.00  0  IIJ  0  IIJ 

J.Jii  l.:!ii  oils 

0  II  0,1  J    0.1:1 

:i7o    (i.;i7  0  .„; 

0.  \i\  0  :;.,  0  iH 

0  n|  O.OJ  II  nil 

ii.ol  on:;  o.iij 

j.oii  i.:i7  0  iH 

(III  O.IJ    0.14 

:i-<o    0.41 

0  .,i 

O.Jo  II  :l  1  0  in; 

o.oi  n.nj  0  07 

001  0  0  :  0  0:1 

1.1)11  i.;t.".  o.x:: 

0.1  J  0.1  :i    0. 1.') 

H'.IO     0. 1.', 

0  .,1 

O.J:t  0.:;:;  ii-ii'i 

0  III   O.OJ  oils 

o.oi  O.OI  0.ii:i 

l.s:!  i.:ij  0.7.". 

O.IJ  0.1:1    O.lll 

400     0.  I'.l  0.  I'.i 

0.J7  0  :i-.'  0  Ml 

0  (i|  II  ,ij  O.IIS 

O.OJ  0  Ol'll.OI 

1.70  1  Jii  0  1:7 

0.1:1  0.1  :i   0  17 

4lo    ii..'i ;  0  n, 

0.:iJ  II  :;i  0  ii:| 

0  n|  II  III  (1  (1:1 

O.OJ  (1  n.'i  ii.o.'i 

I..m'  I.J.',  0  ."ill 

0,1:!  0,1:1    0  IS 

4 JO    ii.r.7  0  1:1 

0.:;r.  II  J!i  0  ii'i 

0  III  11(11  II.  Ill 

11,0:;  nni;  o.inl 

1.4 J  l.Jl  11. ".J 

0.1 1  oil    0  Is 

4.'!o  io.i;ii 

0  10 

0.41  0  j^  odj 

(1  III  n  IIJ  0. 1 1 

0.0,;  0  (ii;  0,01; 

l.2>*  i.lc  II  II 

0.  1  1  0.14     II  111 

410  ,o.f,:i 

o.:m 

0.41',  0  Ji.^o  OJ 

o.oi  O.OJ  11. 1 J 

0.01  0.117  0  07 

l.ll|I.Iii  0  :i7 

0.1.1  0.14     o.li) 

4.^o   n.r,i; 

0  :!l 

0.5J  0  Jl'O  II J 

0  01  (1  OJ  0  1:1 

O.OI  0  lis  0  (IS 

1  (10  Id.",  0  :!I 

0.1. '>  0.14'   0.1!) 

4i;o    (i.i'i'.i 

II  .:i 

o.r)7  0  jj:ii  hj 

0  o|  0  IIJ  n  1 1 

0  0.',  0  (111  0  o'l 

II. S7  O.lis  (1  J.", 

0.  li:  0, 1 1    0  JII 

470    0.7 10  j'.i 

o.r.:!  0  Jl  11 11:1 

n  oj'  (Mi:!  0.  I.'i 

no.',  0  III  0,1(1 

(1.7  1  II.  11 J  0.  Ill 

0.1(1  0   14     II  JO 

4>'0     0.7:!  0  J-. 

O.r.x  II  1'.' :l 

0  IIJ  0  n:;  (1  hi 

II  (IC  0    1(1   0   1 1 

n.ni  OS.',  n  1  1 

0  in  0.14    II, JO 

4'lO  |0.7.'>  o.JI 

0.71  0.17  0  III 

0  (IJ  0  01  0  ir, 

0  07  11.11  0.1  J 

II.. 'HI  0.7"^  o.lo 

0  in  o.i::   ojo 

Ttoo  'n.7r,|ii  ji 

0.7M  0  ).•,  0  o.'i 

0  (IJ  001  0  17 

0.07  0,1  J  0  I'l 

0:111  0.72  0.07 

O.Kl  0.1:1   0  'JO 

.Mo    (1.77  0  I'.l 

O.si  0.1  1  0  111; 

0  (l:;  0  (I.'i  0   1  s 

0  (17  II  1:1  Oil 

11:10  0  (l.'i  11.01 

•  III!  O.l:!    II.JO 

f>Jo    0.7'*  0. 17 

O.MS    II    1  ■_,    II  IIS 

on;  o.oi;  (I  H 

0  (is  0.1 1  0.1 1 

0  JJ  o.^s  II  nj 

0  in  0.1:1   o.lll 

.''i:lil    0.7s  0.  ir. 

O.'.i:!  0.  Ill  0  o'.l 

0  0:;  0  (17  0  IS 

0  (Is  olio  i.'i 

n,  l:'i  0  .',j  n  no 

oin  •I.l  J   •Mil 

■Mo 

0.7H 

0.14 

0.'.I7  O.O'.I  0  10 

0  III  (Ills  0  111 

0,(|s    0    1.1   0    1(1 

n  Oil  0.4."i  0.00 

o.in  O.IJ   0. Is 

f.AO 

0.77 

0  i:t 

1.00  11  (Is  0   1  J 

0  III  0  0:1  0.  Ill 

ii.o;i  oh;  0  17 

0  ll.'i  0,  III  II  00 

o.I.'i  0.12   0  IS 

,'ir,o 

0,7 1', 

0  ct 

1.01  II  117  0  1 1 

0  01  II  oil  0  111 

0  (111  on;  0  17 

0  IIJ  0  :il  II  iiJ 

o.ir.  •l.ll    0.17 

r.70 

0.71 

0  ij 

1  .or.  0  111;  0  1 1; 

o.ii.'i   0.  1(1  II.  Ill 

0.011  0  17  (1  |s 

0,00  0  Jll  0  0| 

ii.i."i  on    0  111 

.^so 

0.7  J 

0  IJ 

JiH  no.',  0  1  s 

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0  Id  0.17  0  |s 

(1.0(1  0  j:,  II  (17 

0.1 1  0.11    01.'. 

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0.70 

0.1  J 

1.10  0  ill  0  I'.l 

0  0(1  11.  1 J  0   111 

0    Id    0    1  s    0    111 

O.IIJ  o.JI  II  10 

0.1  1  II  in    n  14 

•',00 

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o.Oi'  II,  ::i  0.  is 

0     III    0    1  s    (I,  111 

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—144.01 

—211.70 

1  S50 

_ls;!..',7 

105.77  -^'"'^ 

ls:ir. 

r."'.i.4.'. 

is  1.411 

1  HHt) 

It  2.  so 

212.45 

isni 

I72-25  +  "  ■!  = 

Is:i7 

I'.is.:l2 

ls2.(t(i 

|s«7 

lll.7d 

21  a.  12 

|s7o 

ir.o.oi     "  .'' 

202.01     '"'7 
20s.  40     '■   '^ 

ls;!s 

I'.a.ls 

1SL>.(14 

MSS 

llO.dl 

2la70 

Issll 

110.(15     "  -0 

1  s;t>) 

liMl.O.'. 

l.s;{.22 

l«8i» 

iau.6i 

214.4(1 

Is'.iO 

ias.:in    "■-'■ 

ii.j;, 

215.  la  ;;[M 

Hio 

—  r.M.di 

— Ma.sn 

isnn 

— lat^.ao 

—215.1:! 

1000 

-'-^■"'' .,. ,« 

—221.0,')     , 
.j^s.irj— "  00 

Hll 

l'.i:i.7.s 

Ist.liS 

ISO  I 

la7.27 

215.-0 

1010 

115.0s+"''^ 

IMii 

1!I2.('.I 

|s|  Oil 

IS02 

lad.  11 

1     2111.4s 

I'.IJO 

10IS4      "I, 

2,ad!ii     "  ■' 

lsi:i 

I'.tl.M 

1         lS,'..,'i.'( 

iHoa 

la.'..  02 

217.17 

lo::o 

oa.7.-.    ""■' 

2411.4(1     '   '^ 

1H4I 

i'.to.a7 

ls(;.i:i 

1804 

laa.bu 

217. 8t 

1010 

h2.72     "  ''■' 
lo.ii.S 

251.00     "■>■* 
1  li 

iH4r) 

—  1SII.24 

—  ISO. 72 

ISO.'') 

— ia2.77 

! 21^.52 

1050 

—  71.71  ^ 

(;o  ^<..  +IO.OJ 

40.07     '°-^l 
;!o  10     '°"'^ 

— 25S.72     ,  „ 
2dd.d|-7«'> 
274.d,s     ^°7 

2s2.oa    «;; 
.joi.a5    III 

lM4r> 

1      \HH.\i\ 

|s7.al 

IsOd 

;    lai.d,'. 

210.20 

\<MU) 

1S47 
1H4K 

lHr,.ft7 

Is7  110 

1      |ss..Mi 

1S07 

Is'.lS 

:    iao.,'>a 

120.40 

210. ss 
220.57 

1070 
10-0 

1M49 

1S4.70 

1      Isll.oil 

IsOO 

12M.28 

221.2(1 

1 

1000 

2H.40     '°'° 

lu.Ol 

1850 

1 

—  ISO.  (10 

i 

1  000 

-127. Id 

—221.05 

2000 

—   17.s-< 

— 2»il,95 

N 

UK. — 'I'llf 

viiliii  .s  III' 

('•  •-■.!) 

itml  (I'.c. 
XVI 

1 )  iiiii^l  l> 
II  iiii,|  .\ 

•    iMki   11 

V  1 1  li. 

tV.Plll    iililv  •iiir  (,f   1 

II'  Uvii  tallies 

Til  K   Oil  II  IT  OF    I'  U  A  N  VH. 


271 


I 


Vi'iir. 

T 

Alll.K   XVII'.- 

—  ^  'tmvlnih  il. 

(.•.«.8) 

(.•..•.9) 

(W.-..3) 

(ii.r.Si 

(^.r.O)  (p.n.l) 

1 

((.'•.I) 

{(..«.i) 

()..'•.  a* 

(p.«.8) 

(tcS) 

If 

It 

I'llKI 

l.-.lii 

— I'lL'.s:,  , 

IJI.OI     "" 

— in.iio 
10  :i,-. 

—11.70 
ti.711 

.f.|;i71)_  Mill 

liliilt       ss;, 

+2.1(11 
i^J.'.s 

;is2 

111.'. 

_  Ofi 
tic. 

—  NO 

mi 

l.ll'O 

L.L'.oo     °  '* 

I2.-..0.;    "" 

Ill  :il 

11.70 

l:i.'.ii       Ks| 

2I.-.I 

:is2 

172 

01; 

HI 

l.MO 

l.M.f.T     "]\ 

1 ;!.-..  lit     "  '1 

lO.-JH 

(l.tiii 

l:;iii       H7s 

JO  17 

:isi 

170 

!M1 

n\ 

i:.»(i 

I„1.I3     °| 
0.44 

— i."io.r,!t 

0.  16 

lo.Jt 

<■••'••" 

l:i:is       H77 

1041 

:isi 

Is7 

0(1 

H-i 

I'l.iO 

—1  •.>:.. r.o    ^  ,, 

— loao 

—0.(111 

.fl.'l-js—  S7S 

(  ls;iH 

_:isi 

-10.-. 

—  0(1 

—   M2 

i:>i:u 

i;,oj4+°-»; 
1.0  7S    "•■•' 

llO.ilJ     °«' 
us.  hi;    °'^' 

li;..r,7-°    7 
l-.'r..s5    ° ''^ 

III  Hi 

11.110 

l;ilH       ssj 

17:11 

;isi 

20:1 

01; 

83 

!.•.;(» 

III  |:i 

(1.(111 

1:107           HHC, 

1(12:1 

.•{Ml 

2111 

0(1 

M3 

I.".SII 

l-..;.o»    I]'' 

III  00 

(1  (10 

l-.'07       8!i:i 

I.M:i 

.•IS  2 

21 H 

«(1 

N4 

l,'i'.M) 

\>i\.>i    °'" 

10  O'l 

(1.70 

IJsd       001 

1 4o;i 

3s2 

2-2ii 

0(1 

H4 

0.47 

O.iO 

1 1,1  Ml 

—  1  IS  .'lO 

- 1  •.'«;.  H 

—  Ill  OJ 

— (1.70 

l-IJ7,'._  Oil 

(  I-J02 

_;is:i 

_2:i» 

_  (Id 

Hh 

li.lO 

ll7..,ij  )-"  17 

ijr,.  (;.;-"-'» 

H  OS 

(1,70 

IJdl        Oj:i 

llso 

:isi 

212 

0(1 

Nil 

ll'.JII 

II7,M     °  ••* 

JL'C.HO     "-'1 

O.lll 

(1,71 

1  J.->:1        0:17 

10(17 

:is-. 

2.1(1 

)l(i 

Hti 

|i;:l(l 

lir.oii    °  t« 

IJ7.l:i    °-' 

O.lll 

(1.71 

1 J  ( 1        0.".  1 

o.^»     Use. 

•J.-is 

0(1 

n^ 

IC.IU 

1111.18     °   »'^ 
0  40 

i-j7.;h   " -; 
0.  ](i 

11.  s7 

(1.7  J 

UMii        07  J 

H4(t      .'{ss 

2(>(l 

0(> 

87 

I  ti.'.O 

— 1  ir,.<M\ 

— l-.»7.fi» 

—  ft.s:j 

-fi.7:i 

j  l-JIO 00 J 

t-   72.'i— S-^O 

—274 

_  0(1 

MS 

n;<;o 

ll.-,..Ml +"■»'' 

IJ7.'.l|-°  '7 

!l.70 

•1  71 

Il'os      Hill 

(110     :ioi 

2S2 

0(1 

Hit 

UTo 

11;-.  00     °''° 

I-.N.IO     '■'■"< 

!i.7:. 

(1.7.'> 

11117      10  :> 

40.')     :io2 

20(1 

oil 

MO 

|r,>o 

iii.rio    ° ''° 

l.'S.  Ill      °   ?'' 

'.1.71 

(1.7(1 

11".".      Iiiill 

.•170,     .T.il 

20s 

m 

00 

ir.'.iii 

111.00    °  ■;" 

°  5' 

1-js  su     °  1' 

li.r.7 

(1.77 

1171      looj 

2(12j     aod 

MIHI 

m 

110 

17IM1 

-li:!.r.)  , 

-KMlId 

—  !Mi:i 

—(1.7s 

1  llii:i_ll-.':i 

f.  145  _:ios 

-Jill 

—  07 

—   01 

ITHt 

1U!IS+°S' 

l:>l..ir.-°.V' 

!i.:.ii 

(1  711 

ll.vj      II.-..-. 

(-     27      41HI 

.•1J:| 

07 

02 

17. '11 

11-.'  17     °'^' 

Il'0  71I     °  '» 

ii.r.:> 

(l.HO 

nil      l|s;i 

_    0-.'      40:1 

:i:il 

07 

02 

1710 

111.  111!     ^'•'^ 

1:10.  n    °  «> 

(i.ril 

(l.sl 

11:10      I'j'jii 

21 J      4ii.% 

:M!» 

07 

0:1 

1710 

III. It     °^^- 
05' 

l:io.;.l     °  57 
o-,?7 

'.1.17 

(•...s:i 

11  111      IJill 

;i:i:i     40s 

:us 

OS 

03 

17.>ii 

-1  10.0:1  , 

— l:i(»,ss 

—  !l.  13 

— (i.si 

\  llos_|::oi  _  4.M  —III 

_:i.-.(! 

—   OH 

—  04 

l7i;o 

lio.ll+°-''- 

|:!l  -.'T-"'" 

!i.;ii> 

(i.sr, 

1007     i;iiii'     .'.7.'.     411 

IKl.-i 

OS 

0.'. 

1770 

1  iO  ss      '   ■!« 

|:il.r,7    °  '° 

0.,^^ 

(1..S7 

Klsil       1:101 

(10(1      417 

:i7:i 

us 

0;"> 

17  Ml 

1:10,  :t.i    °>- 

|:!-.>iis    °  •' 

!i.:il 

(l.sll 

I1171       11:7 

SI  7      4  JO 

.■|s2 

00 

0(1 

1700 

l:is.s.|    "  vi 
0.5  J 

l:L'..-.o    °-t* 
04.1 

O.i'T 

n.oo 

lllll.l        llsil 

y.is     4JI 

300 

00 

0(1 

|S00 

.-l:is.:ii 

~l:ia.o:i 

—  51. '.V! 

— (i.O-J 

fl(r,i_l.''.:i(i— 10:10—4  •-'7 

—30!) 

—  00 

—  07 

I'^IO 

1:17.70+°' 

1:1:1:17 -°" 

11.111 

(i.o:i 

Mill       l.'.ss       llso       1:11 

4  OS 

00 

OS 

1-JO 

1.17.  •jr.    "■^■' 

1:1:1. sj   " '; 

0. 1  1 

(l.O.'i 

|ii:|ii      lill'J      l:liil       |:!.'> 

41(> 

00 

OS 

lv;o 

'■'•••'•    oT. 

i:ii.-.'s  ^  ;^- 

0.41; 

0.10 

(i.iin 

III  111      MOO 

1 IJJ    4:10 

*•>■> 

ion 

1^|0 

.:;.i.o,    °|] 

—  1  :!.■>. no  . 
l:i.V17+°''' 

ii.ih; 

(l.lis 

liiils       17.'.>> 

l.it     4i:i 

4:13 

loo 

00 

1  x.iO 

_  ll.O'J 

—7  (10 

)-  007  _isio_ir,r,:,  _4t7 

—413 

—loo 

—  loo 

|>r,i) 

|:i.-..7.-.-"-" 

h.OS 

7.0-J 

llsil       lss| 

17S.1      4..I 

4.'.(l 

loo 

llll 

1^70 

i:!i.i;.-.    °t 

l.itl:i     ^ 

,:o;.-.il    -:'. 

K.m 

7.01 

117.".      10l."i 

100(1      4.'.(1 

4.MI 

lol 

Kil 

1  «sll 

|:ii;.7s     "    - 

s.oo 

7.07 

Olll         L'lll-.' 

•J0J7      4 '11 

4117 

l(i| 

loj 

1  >;io 

''^"^'  ::v 

.:.7.:.l    -;:| 

8.HC, 

7.00 

0..:i      -josl 

2117      41.11 

4  70 

KU 

1(12 

r.iiio 

-l:!:i  01)  , 

—  |:!7.sj 

—  8.S-J 

— 7.I-.' 

1    ii|j_-.'l.-.l  _'j-.'i;7— 471 

_|S1 

-loj 

—  1113 

I'.llO 

l:i-.'.:.7+°-^; 

|:is  lo'--^  S5 
l:!s.oi;    °S^> 

H.7S 

7.1.-. 

0:1      •.'l'l':i     •j:ts(i     4711 

40J 

liij 

llll 

lOJO 

l:iJiMi    °S' 

H,7I 

7,17 

Oill      •.'■.'07      •-'.'lO.'.      4'l 

.MIO 

10:1 

lli« 

lo;;o 

i.;l..v.    °>' 

l::o;,:i    "" 
110.11     "^'^ 

s  70 

7  I'll 

Oln      -.'.i:-.'      -JUi':!      4»(1 

.MIO 

1(13 

1(1.-. 

1 010 

1:11.0 J    °5' 

H.r.ii 

7.1' J 

Olio      L'llO      27 11      41''J 

.'.17 

10:! 

III.') 

0.50 

0  50 

r.i.'iO 

—  i:io,-.i  , 

— H(l  70        , 
I4l.:i|-"^" 

—  S.C.J 

— 7.  J.'. 

(-   Hso '.>,"i'.'0 2S."iO 107 

— :,•>:, 

—  101 

—  IlKl 

r.iiio 

Mil  111+°  5° 

s,;is 

7.'js 

s70      'Jdll       207(1      :'.(i.! 

:>.;:\ 

lot 

inn 

ri7ii 

lj!i:>.-.    °-»'' 

111.0:1    "  ■- 

S..M 

7  :il 

sr.s      •.Till.'.      ;i!iii-.>      fiiiii 

.Ml 

10.-. 

1(17 

I'.ISd 

ljo.0.1    °-»'' 

M-.'..Ml     "   '•' 

li;t.-jo    °fi 

S  ."lO 

7  :il 

n;.s      •J7"^0      :!-Jii7      .M.". 

.'■..'.(I 

1    1(1.". 

1117 

I'.l'.lll 

l-.-s.,-,7    °  '•' 

.s   111 

1  .  ->  1 

SIS      -jsi;;      :i:;:..|      :,j| 

.'..•.s 

1     1(1(1 

111.. 

o.4,S 

o.^5 

■JIM  III 

—  IJS  0!) 

-in  s:, 

—  s  {'2 

-7   I'l 

I     s;is__-."i.-,i,  -.;i(:;i      ,-,j: 

.Mir, 

—  Kill 

_|iiH 

/ 


^%. 


%>  t^ 


IMAGE  EVALUATION 
TEST  TARGET  (MT-S) 


1.0 


I.I 


1.25 


m 

m 


IIM    IIIIL2.5 


'-  IIM  IIIIIZ2 
2.0 


lAO 


1 1.8 

U    III  1.6 


Photographic 

Sciences 
Corporation 


23  WEST  MAIN  bTRflE' 

WEBSTER,  N.Y.  )45t:0 

(716)  873-4503 


4 


4< 


.4^ 


^^ 


\\ 


^v 


O^ 


#.^ 


'^^^ 


^ 
^o^ 


f/j 


i 


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o'^ 


<!^^- 


^ 


272 


THE   ORBIT   OP   URANUS. 


I 


TA15LK  XVIII Reduction  to  ths  Ecliitio.  Abqument  u. 


ISO 
181 
1«2 
1S3 
184 

185 
ISC, 
187 
188 


9  189 


20 
21 
22 
23 
24 

25 

2(> 
27 

28 
29 


32 
33 
34 

35 
3(i 
37 
38 
39 

40 
41 
42 
43 
44 

45 


190 
191 
192 
193 
194 

195 

19(> 
197 
198 


19  199 


200 
201 
202 
203 
204 

205 
20() 
207 
208 
209 


30  210 

31  21 1 


212 
213 
214 

215 
2  III 
217 
218 
219 

220 
221 
222 
223 
224 

225 


R 


10.00 
9.(>7' 
9.35 
9.02 
8.70 

8.37 

8.05' 

7.73 

7.42 

7.10 

0.80 

C.49- 

0.19 

5.89 

6.00 

5.32 

5.04" 

4.70 

4.49 

4.23 

3.98 

3.73- 

3.49 

3.2(; 

3.04 

2.82 

2.(;2- 

2.42 
2.23 
2.05 

1.8S 

1.73- 

1.58 

1,44 

1.31 

1.20 
l.Olt 
0  99 
0.91 
0.»4 


-°.33 
0.32 

°-33 
0.32 

°-33 

-0.32 
0.32 
0.31 
0.32 
0.30 

.0.31 
0.30 
0.30 
0.29 
0.28 

-0.28 
0.2S 
0.27 
0.26 
0.25 

-0.2s 
0.24 
0.23 
0.22 
0.22 

-0.20 
0.20 
0.19 
0.18 
0.17 

-0.1 5 
0.15 
0.14 
o  13 

O.  II 

— O.II 
O.IO 

0.08 

0.07 
0.07 

0.77 

0.72 -°°5 

0.08  °'°-^ 
0.03 

O.OI 
0.01 


0.05 
0.04 

0.03 


50 
51 
52 
53 


55 
50 
57 
58 
59 

00 
01 
02 
03 
04 

05 
00 
07 
08 
09 

70 
71 
72 
73 
74 

75 

70 
77 
78 


45   225 
40  I  220 

47  ;  227 

48  I  228 

49  229 


54  234  1.09 


230 
231 
232 
233 


235 

230 
2371 
238  j 
239 

240 
241 
242 
243 
244 

245 
240 
247 
248 
249 

250 
251 
252 
253 
254 

255 
250 
257 

258  I 
2591 

200 

201  I 

202  i 


R 


0.03  , 
0.04+°°' 

O.OI 


0.05 
0.08 
0.72 


0.03 
0.04 
0.0; 


2:n+°°v 

0.99  °°8 


1.20 

1.31+°" 
1.44  °'-5 
1.58  °'-* 
1.73  °'5 
0.15 

1.88 

2.05+°- '7 
„  .,.,     0.18 

240     °'9 

2.02     °-° 
0.20 

2.82  , 

3.04+°-^-' 

3.20 

3.49 

3.73 


0.23 
0.24 


3.98  , 
4.23+° 
4.49     ° 
4.70     ° 
5.04 


0.28 


5.32 
5.00 
5.89 
0.19 
0.49 


+0 
o 
o 
o 
o 


S4 

203 
204 

85 

205 

80 

200 

87 

207 

88 

208 

89 

209 

90 

270 

0,80 

7.10+°- 

7,42     °- 

7.73     °- 

8.05     °- 
o. 

8.37  , 
8.70+°- 
9.02     °- 
9.35     °- 
9.07     °' 


90 
91 
92 
93 
94 

95 
90 
97 
98 
99 


270 
271 
272 
273 

274  j 

275' 
270  \ 
2771 
278 
279  I 


110 
111 
112 
113 
114 

115 

110 
117 

118 
119 

120 
121 
122 
123 
124 


125 

3° 

120 

32 

127 

31 

128 

3-! 

129 

32 

130 

33 

131 

3^ 

132 

,>,.•, 

133 

3-! 

134 

33 

R 


10.00  , 

10.33+°-" 

10.05  °-3^ 

10.98  °'" 

11.30  °-3^ 
0-33 

11.03  , 
11.95+°- 3^ 


12.27 
12,58 
12.90 


0.32 
0.31 
0.32 
0.30 


1001  280 : 

101 i  281 
102:  282 

103  1  283 

104  I  284, 

105  285: 
100!  280 ! 
107^  2S7 

lOS  2S8 
109  289 


13.20 

13.51 +°-3' 
13.81 
14.11 
14,40 


0.30 
0.30 
o,  29 
0,28 


290 

291  I 
292 
293 
294: 

295 
2911 
297  i 
298 
299 

300 
301 
302 
303 
304 
I 
305 
3(10 
307 
30S 
309 

310 
311 
312 
313 
314 


14.08  ,  „ 
14,90+°-=^ 
15,24  °-^'^ 
15.51  °%7 

15.77  °'^ 

0.25 

10.02  , 

10.27 +°-'5 

10.51  °-^^ 
10.74  °-'3 

10.90  °" 
0.22 


17.18 

n,38 
17.58 
17.77 
17.95 


4-0.20 
0.20 

o.  19 
0.18 

0.17 

18.12  , 

18  27+°-'5 

19.42  °'5 

1>^<'«  o.M 

18,80 

18.91+°" 
19,01   °-'° 

19.09  °-°'^ 

19.10  °°7 
0.07 

19.23 

]<l,.2,s+°°5 

19,32 


19.35 
19.30 


0,04 
0,03 

O.OI 
O.OI 


10.00 


135.315  19.37 


135  315 
130  310 
137,  317 
1381  318 
1391  319 

140'  320 

141  !  321 

142  322 
143'  323 

144  1  324 

145  ;  325 
140  j  320 

147  327 

148  328 

149  329 

1501  330 

151  331 

152  332 

153  333 
154 ,  334 

155  335 


150 
157 
158 


330 
337 
338 


170 
171 
172 
173 
174 

175 

170 
177 

17s 
179 

ISO 


159  339 

100  [  340 

101  j  311 
102]  342 
103,  343 
104'  344 

105  345 
100,  340 
107:  347 

108  348 

109  349 


350 
351 
352 
353 
354 

355 
350 
357 
35H 
359 

300 


R 


19.37 

19.30" 

19.35 

19.32 

19.28 

19.23 
19.10 
19.09 
19.01 
18.91 

18.80 

18.09' 

18.50 

18.42 

18.27 

18.12 

17,95" 

17.77 

17.58 

17.38 

17.18 

10.90' 

10.74 

10.51 

10.27 

10.02 

15.77" 

15.51 

15.24 

14.90 


-O.OI 
O.OI 

0.03 

0.04 

0.0s 

—0.07 
0.07 

0.08 

O.IO 
O.II 

-O.II 

0.13 

0.14 

0.15 

-0.17 
0.18 

0.19 
0.20 
0.20 

-0.22 

0.22 
0.23 
0.24 
0.25 

-0.25 
0.26 
0.27 
0.28 
0.28 


14,40-°f 

0.30 
0.30 
0-3I 


13.81 
13.51 


13.20 

12.90" 

12,58 

12,27 

11.95 


-0,30 
0.32 
0.31 
0.32 
0,32 


11,03 

11.30" 

-0'33 

10,98 

0.32 

10.05 

°-33 

10.33 

0.32 

0-33 

10.00 

i  I 


f.1  I 
|,l 


THE   ORBIT   OF   URANUS 

213 

TABLE  XIX.— 

I'lUNCIl 

AL  Tku.m  of  the  Latitude.     A 

lUiLMENT  M. 

u 

(3 

U 

3 

u 

(3 

180° 

/           /' 

360° 

190° 

t           n 

350° 

200' 

/                        " 

340° 

0° 

0      0.00 

8.09 

180° 

10' 

8     2.82 

7.96 
7.96 

170° 

20° 

15  50.98     , 
15  58.58     I 

60 

160° 

10' 

0     8.09 

50' 

10' 

8  10.78 

50' 

10' 

e  t\ 

50' 

20 

0    10.18 

8  09 

40 

20 

8  18.74 

40 

20 

16     6.17     III 
16  13.75     I'l^. 
16  21.32     l\l 
16  28.88      '-^S: 

16  36.43       ' 

16   43,98     l'l\ 
16  51.52     l\-l 

40 

30 

0    24.27 

8.09 

30 

30 

8  20.09 

7-95 

30 

80 

30 

40 

0    32.35 

8.08 

20 

40 

8  34.04 

7-95 

20 

40     i 

20 

50 
1° 

0   40.44 
0    48.53 

8.09 
8.09 

8.09 
8.08 

10 
179° 

50 
11° 

8  42.59 
8  50.54 

7-95 
7'95 
7-94 
7-93 

10 
169° 

60 
21° 

10 
159° 

10' 
20 

0  5(5.02 

1  4.70 

50' 

40 

10' 
20 

8  58.48 

9  0.41 

50' 
40 

10' 

20   ; 

50' 
40 

30 

1    12.79 

8.09 

30 

30 

9  14.34 

7'93 
7.92 
7.92 
7.91 

30 

30 

16  59.05     ' 

•SI 
•5° 

30 

40 

1    20.87 

8.08 

20 

40 

9  22.26 

20 

40     ; 

17     6.57     ; 

20 

50 

1    28.9G 

8.09 
8.0S 

10 

50 

9  30.18 

10 

50     . 

17   14.08     ' 

10 

2° 
10' 

1    37.04 
1    45.12 

8.08 
8.08 
8.08 
8.08 
8.08 
8.08 

178° 

50' 

12° 

10' 

9  38.09 
9  45.99 

7.90 
7.90 
7.90 
7.89 
7.89 
7.89 

168° 

50' 

22'  ; 

10' 

17  21.58 
17  29.07     ; 

•49 

.48 
.48 

•47 
.46 

1,58° 

50' 

20 
30 
40 

1  53.20 

2  1.28 
2      9. 30 

40 
30 
20 

20 
30 
40 

9  53. ao 

10     1.79 
10     9.68 

40 
30 
20 

20 

30     i 

40    ; 

17   36.55     I 
17   44.03     ; 
17  51.50     ' 

40 
30 
20 

50     , 

2    17.44 

10 

50 

10  17.57 

10 

50     ' 

17  58.90     ; 

•45 

10 

3" 

10' 

2    25.52 
2    33.00 

8.08 

177= 

50' 

13° 

10' 

10  25.40 
10  33.34 

7.88 
7.87 

167= 

50' 

23°    ' 

10' 

18     fi.41 
18  13.85     I 

•44 
•43 
.42 
.41 
.41 
.40 

•39 
•37 
•37 
■ii 
•35 
•33 

•33 
•31 

•  ^0 

157° 

50' 

20 

2    41.07 

8.07 

40 

20 

10  41.21 

40 

20 

13  21.28     I 

40 

30 

2    49.75 

8.08 

30 

30 

10  49.08 

7.87 
7.S6 

7. SO 
7-85 

7.8.=; 
7,84 
7-8.3 
7-8,3 

30 

30 

18  28.70     ' 

30 

40 

2    57.82 

8.07 

20 

411 

10  56.94 

20 

40     1 

18  36.11     ' 

20 

60 
4° 

3     5.89 
3    13.90 

8.07 
8.07 

10 
176° 

50 
14° 

11     4.80 
11   12.65 

10 
166° 

50     1 
24'   1 

18  43.52     ; 
18  50.92     , 

10 
156° 

10' 

3    22.03 

8.07 

50' 

10' 

11   20.50 

50' 

10' 

18  58.31     i 

50' 

20 

3    30.09 

8.c0 

•10 

20 

11   2S.34 

40 

20 

19     5.68     i 

40 

30 

3    3S.10 

8.07 

30 

30 

11   36.17 

30 

30 

19  13.05    -i 

30 

40 

3    40.22 

S.06 

20 

40 

11   44.00 

20 

40 

19  20.40     i 

20 

50 
5' 

3  54.28 

4  2.34 

8.06 
8.0O 

8.05 
8.06 

10 
175° 

50 
15' 

11  51.82 
U  59.63 

,.h2 

7. Si 

7.  So 
7.S1 
7.80 

7-79 
7.  78 

7-79 
7.7;! 
7-77 
7-75 
7-75 
7-74 
7-74 

7-7.3 

10 
165° 

50 
25' 

19  27.75     ; 
19  35.08     ! 

10 
155° 

10' 

4    10.39 

50' 

10' 

12     7.43 

50' 

lo' 

19  42.41      i 

50' 

20 

4    18.45 

40 

20 

12  15.24 

40 

20 

19  49,72     i 

40 

30 

4    211.50 

8.05 

30 

30 

12  23.04 

30 

30 

19  57.02     i 

0 

•30 
.29 

•  27 

.20 

30 

40 

4    34.50 

8.o(j 

20 

40 

12  30. S3 

20 

40 

20     4.32     i 

20 

50 
6" 

4    42.61 
4   50.05 

8. 05 
8.04 

8.04 

10 
174° 

50 
16° 

12  38.61 
12  40.40 

10 
164° 

50 
26° 

20 11.61  ; 

20   18.88     . 

10 
154° 

10' 

4  5s.(;'.i 

50' 

10' 

12  54.18 

50' 

10' 

20  26.14     . 

.20 

50' 

20 

5     (;.73 

8. 04 
8.03 
8.0J 

40 

20 

13     1.95 

40 

20 

20  33.40     . 

f.24 

40 

30 

5    14.70 

30 

30 

13     9.70 

30 

31) 

20  40.64     . 

J.  24 

30 

40 

5    22.79 

20 

40 

13  17.45 

20 

40 

20  47. 8S 

h  22 

20 

50 

5    30. S2 

8.0J 
8.0J 

8- 0,1 

8.02 

10 

50 

13  25.19 

10 

50 

20  65.10 

?-2I 

10 

T 

5    3.-i.s5 

173° 

17° 

13  32.93 

163° 

27° 

21     2.31 

7.20 

153° 

10' 

5    41). US 

50' 

10' 

13  40.06 

."lO' 

10' 

21     9.51 

7.19 
7.IS 

50' 

20 

5    54.90 

40 

20 

13  48.39 

7-7.1 
7-7^ 
7-/1 
7.70 
7.O9 

7-^19 
7. 08 

7-O7 
7.O7 
7.00 
7.(>5 

7-65 
7,04 

7-f',3 
7.O2 
7.61 

7.O0 

40 

20 

21   16.70 

40 

30 

0      2.92 

8.C2 

S.o_' 

30 

30 

13  56.11 

30 

30 

£1   23.  S8 

7-17 
7.1O 

30 

40 

(i    10.94 

20 

40 

114     3.S2 

20 

40 

21   31.05 

20 

50 

0    IS.  95 

8. 01 
8.01 

10 

5') 

'\  14  11.52 

10 

50 

21  38.21 

1 

7.14 

10 

8° 

10' 

0    2l!.9(! 
0    31.97 

S.oi 

172° 

50' 

18' 

10' 

14  19.21 
It  26.90 

162' 

50' 

28' 

10' 

:;l   45.35 
21  52.49 

7.14 
■?    I  2 

162° 

50' 

20 

(i    42.97 

8.00 

40 

20 

14  34.58 

40 

20 

21   59.61 

■7    13 

40 

30 

1)    50.97 

8.00 

30 

30 

'  14  42.25 

30 

30 

22     0.73 

/•  '-^ 

30 

40 

(i    5S.97 

8.00 

20 

40 

14  49.92 

20 

40 

22  13.83 

7.10 
7,09 
7,08 

20 

50 

7      (•,.90 

7-9'; 

8.00 

10 

50 

14  57.58 

10 

50 

22  20.92- 

10 

9° 

7    14.90 

171° 

19' 

15     5.23 

161° 

29° 

22  28.00 

7,07 
"7   0^ 

151° 

10' 

7    22.95 

7-99 

50' 

10' 

15   12.S8 

50' 

10' 

22  35.07 

5(i' 

20 

7    .'i(l.9;i 

7.9S 

40 

20 

15  20.52 

40 

20 

22  42.12 

7-05 

7-°3 
701 

40 

30 

7    3S.91 

7.98 

30 

30 

15   28.15 

30 

30 

22  49.17 

30 

40 

7    4(1. S8 

7-97 

20 

40 

15  35.77 

20 

40 

22  56.20 

20 

50 

7    54.85 

7.97 

10 

50 

15  43.38 

10 

50 

23     3.23 

/  -^.1 
7.01 

!          10 

10° 

8      2.S2 

7-97 

170° 

20° 

15  50.98 

160' 

30° 

23   10.24 

150° 

190° 

!i 

350° 

■u 

200° 

/5 

340' 

u 

210° 

a 

830° 

u 

35       July,  1873. 

271 


THE    ORBIT    OF   TTRaNUS, 


TABLE 

XIX,  Alio.  M. — Cuiilin 

ued. 

u 

'        3 



u 

P 

u 

3 

210^ 

/            1/ 

330" 

220" 

/            »/ 

320" 

230° 

/            V 

310° 

30" 

2;3  10.24 

150" 

40" 

29  47.27 

6  t9 
6., 7 
6.16 
6.14 
6.13 

140 

50" 

35  29.99 

130° 

10' 

23  17.24 

7.00 
6.99 
6.1,7 
6.97 
(>')5 

50' 

10' 

29  53.40 

50' 

10' 

35  35.18 

5- 19 

50' 

20 

23  24.23 
23  31.20 

40 
30 

20 
30 

29  59.03 

30  5.79 

40 
30 

20 
3(' 

35   40.35 
35   45.50 

5'7 

5-'5 

40 
30 

40 

23  3S.17 

20 

40 

30   11.93 

20 

4j 

35   50.03 

5- '3 

20 

r)0 

23  45  12 

10 

50 

30  13.06 

10 

60 

35  55.75 

512 

10 

6.94 

6.  II 

5.10 

31° 

23  52.00 

6-93 
6.91 
0.90 
6.89 
6.88 
0.86 

149" 

41" 

30  24.11 

6.10 
6.08 
6.06 
6.05 
6.03 
6.02 

139° 

51° 

30     0.85 

5.08 

129° 

10' 

23  5S.!)i) 

50' 

10' 

30  30.27 

50' 

10' 

3(!     5.93 

50' 

20 

ao 

40 
50 

24     5.!»0 
24  12.80 
24  ll).(i!) 
24  20.57 

40 
30 
20 
10 

20 
30 
40 
50 

30  3f;.35 
30  42.41 
30  48.40 
30  54.49 

40 
30 
20 
10 

20 
30 
40 
60 

30   11.00 
30   10.05 
30  21.07 
30   20.08 

5-07 
5-°5 
5.02 
5.01 
4-99 

40 
30 
20 
10 

32' 

24  33.43 

6.85 
6.84 
6.83 

148° 

42° 

31     0.51 

138° 

52" 

30  31.07 

128° 

10' 

21  40.28 

50' 

10' 

31      0.51 

6.00 

50' 

10' 

30   30.04 

4-97 

50' 

20 

24  47.12 

40 

20 

31    12.50 

5-99 

40 

20 

30   40.99 

4-95 

40 

30 

24  53.05 

30 

80 

31    18.47 

5-97 

30 

30 

30  45.93 

4.94 

30 

40 

2.-I     0.77 

6.82 

20 

40 

31   24.43 

5-96 

20 

40 

30   50.84 

4.91 

20 

M 

25     7.57 

6.80 

10 

50 

31  30.37 

5-94 

10 

50 

30  55.73 

4. 89 

10 

33= 

25  14.3r, 

0.79 

6.7s 
6,76 

6-75 
6.74 
6.7.' 

147" 

43" 

31   30.29 

5-92 

137° 

53" 

37     0.01 

4.88 

127° 

10' 

25  21.14 

50' 

10' 

31   42.20 

5-91 

50' 

10' 

37     6.47 

4.86 

50' 

20 

25  27.90 

40 

20 

31   48.09 

S-«9 

40 

20 

37   10.31 

4-84 

40 

;{0 

25  34.05 

30 

30 

31  53.90 

S-«7 

30 

30 

37   15.13 

4. 82 

30 

40 

25  41.39 

20 

40 

31  59.82 

5. 86 

20 

40 

37   19.93 

4.80 

20 

CO 

25  48.11 

10 

50 

32     6.00 

5-84 

10 

60 

37  24.72 

4-79 

10 

34" 

25  54.82 

6.71 

6.70 
6.68 
6.68 
6.66 
6.65 
6.6,5 

6.62 

146° 

44" 

32   11.49 

5.81 

136" 

54" 

37  29.48 

4.76 

126° 

10' 

■2C,     1.52 

50' 

10' 

32   17.30 

50' 

10' 

37  34.22 

4-74 

50' 

20 

;io 

2()     8.20 
.'<i   14.88 

40 

30 

20 
30 

32  23.09 
32  28.87 

5-79 
5-78 
576 

40 
30 

20 
30 

37   38.95 
37  43.00 

4-73 
4-7' 

40 

30 

40 

2r,  21.54 

20 

40 

32  34.03 

20 

40 

37  48.34 

4. 68 

20 

50 
35" 

20  28.19 
20  34.82 

10 
145" 

60 
45" 

32  40.38 
32  40.11 

5-75 
5-73 

10 
135" 

60 
55° 

37  53.01 
37  57.00 

4.67 
4-('5 
4-63 

10 

125" 

10' 

20  41.44 

50' 

10' 

.■12  51.82 

5-7' 

50' 

10' 

38     2.29 

50' 

20 

20  4S.04 

6.60 
6.59 
6.58 
6.56 
6.55 

6-54 
6.52 
6.5. 
6.50 
6.,i8 
6.47 

40 

20 

32  57.51 

5  •''9 
5.68 

40 

20 

3S     fi.OO 

4.61 

40 

:iO 

20  54.03 

30 

30 

33     3.19 

30 

30 

38   11.49 

4-59 

30 

40 

27     1.21 

21) 

40 

33     8.S5 

5.66 
564 
563 
5.61 

20 

40 

3S    10.07 

4-58 

20 

50 

27     7.77 

10 

50 

33   14.49 

10 

50 

38  20.02 

4-55 

10 

36" 

27   14.32 

144" 

46" 

33   20.12 

134" 

56° 

38   25.15 

4-53 

124" 

10' 

27   2(1. 8(i 

50' 

10' 

3;i  25.73 

50' 

10' 

38   29.00 

4-51 

50' 

20 

27  27.38 

40 

20 

33  31.32 

5-.=;9 

5-58 

40 

20 

3S  34.10 

4-5° 

40 

;iO 

27  33.89 

30 

30 

33  30.90 

30 

30 

38  38.03 

4-47 

30 

40 

27  40.39 

20 

40 

33  42.40 

20 

40 

38   43.08 

4-45 

20 

50 

27  40.87 

10 

50 

33  48.00 

5-54 
5o3 

10 

50 

38  47.52 

4-44 
4.41 

10 

37" 

10' 

27  53.34 
27  59.79 

6-45 
6.44 
0.4J 
6.4, 
6.40 
6.58 

6.37 
6-35 

6.3 1 
6.  J9 

6.28 
6.26 
6.25 

6.24 
6.22 
6.20 

143" 

50' 

47^ 

10' 

33  53.53 
33  59.04 

5-51 

133' 

50' 

57" 

10' 

38  51.93 
38   6(!.33 

4.40 

123" 

5o' 

20 

28     0.23 

40 

20 

3t     4.53 

5-49 

40 

20 

39     0.70 

4-37 

40 

;!o 

28  12.05 

31) 

30 

34   10.00 

5-47 

30 

30 

39     5.00 

4-3'^' 

30 

40 

28   10. 01) 

20 

40 

34   15.40 

5-46 

20 

40 

39     9.40 

4.34 

20 

50 

28  25.40 

10 

50 

34  20.!)0 

5-44 

10 

60 

39   13.71 

4-31 

10 

38" 

28  31.84 

142' 

480 

34  20.32 

5-42 

132' 

58" 

39   18.01 

4'.5o 

122" 

10' 

28  38.21 

50' 

10' 

3t  31.72 

5-40 

511' 

10' 

39  22.29 

4,2s 

50' 

20 

28  44.50 

40 

20 

34   37.11 

5-39 

40 

20 

39   20.51 

4- -'5 

4  0 

.'iO 

28  50.89 

30 

30 

34   42.48 

5-37 

30 

30 

39   30.78 

4.24 

30 

40 

28  57.22 

20 

40 

34  47.83 

5-35 

20 

40 

39  34.99 

4.21 

20 

50 

29     3.53 

10 

50 

34  53.10 

5-33 

10 

50 

39   39.19 

4.20 

10 

39" 

29     9.82 

141" 

49' 

34  58.47 

5-31 

131° 

59" 

39  43.30 

417 

121° 

10' 

29  10.10 

50' 

10' 

35     3.77 

5-30 
5.28 
5.26 

50' 

10' 

39   47.52 

4. 16 

50' 

20 

29  22.30 

40 

20 

35     9.05 

40 

20 

39  51.05 

4J,' 

40 

MO 

29  28.01 

30 

30  : 

35   14.31 

30 

30 

39  55.77 

4.12 

30 

41) 

29  34.85 

20 

40     , 

35  19.55 

5*4 

20 

40 

39  59. s 7 

4. 10  1 

20 

50 

29  41.07 

10 

50   ; 

35  24.78 

5-23 
5.21 

10 

50 

40     3.91 

4.07 
4.06 

10 

40' 

29  47.27 

140 

50' 

35  29.09 

130 

60" 

40     8.00 

120" 

9.90' 

? 

320" 

u 

230" 

0 

310' 

u 

240 

0 

300" 
'        1 

r 


THE   ORBIT   OF   URANUS. 


215 


TA15LE  XIX,  Alia.  u.—  CoiUinued. 


u 


240 

60° 

10' 
20 
30 
10 
50 

61° 

10' 
20 
30 

40 
50 

62° 

10' 
20 
30 
40 
50 

63° 

10' 
20 
30 
40 
50 

64° 

10' 
20 
30 
40 
60 

65° 

10' 
20 
30 
40 
50 

66° 

10' 
20 
30 
40 
60 

67° 

10' 
20 
30 
40 
50 

68° 

10' 
20 
30 
40 
50 

69" 

10' 
20 
30 
40 
50 

70° 

260° 


8.00 
12.03 
10.04 
20.04 
24.01 
27.90 

31.89 
3'>..S0 
40  39.09 
40  43.50 
40  47.41 
40  51.24 

40  55.05 

40  5.><.84 

2.00 

0.35 

10.07 

13.78 

17.40 
21.13 
24.77 
2K.39 
31.99 
35. 5G 

39.12 
42.05 
40.17 
49.00 
5.!.  13 


50. 5S 

O.Ol 
3.42 

o.so 
10.17 
13.51 

10.83 

20.13 
23.41 
20.07 
29.91 
42  33.12 
42  30.32 

42  39.49 
42  42.04 
42  4."). 77 
42  4S.S7 
4  2  51.90 
42  55.02 

42  58.00 

43  1.08 
4.08 
7.05 

10.01 
12.94 

15.85 
18.74 
21.00 
24.45 
27.27 
30.00 

32.84 

3 


4- 03 
4.01 
4.00 
3-97 
3-95 
3-93 

391 
3«9 
3-«7 
3-85 
383 
3.S1 

3-79 
3. 76 
3-75 
3-7^ 
371 
3.6S 

367 

3(>4 

3.62 

3.60 

3-57 

356 

3-53 

3-5^ 

3-49 

3-47 

3'4.=; 

3-43 

3- 4 1 

3-3« 

3-37 

3-34 

332 

3i° 

3,28 

3.26 

3-^4 

3-2I 
3.20 

317 


43 
43 
43 
43 

43 
43 
43 
43 
43 
43 

43 


31.=; 
3- '.I 
3.10 

3°9 
3.06 

3°4 
3.02 
3.00 
2.97 
2.96 

2-93 
2.91 

2.89 
2.86 

2.85 
2.82 
2.79 
2.78 


300° 

120" 

50' 
40 
30 
20 
10 

U9° 

50' 
40 
30 
20 
10 

118° 

50' 
40 
30 
20 
10 

U7° 

50' 
40 
30 
20 
10 

116° 

50' 
40 
30 
20 
10 

115° 

50' 
40 
30 
20 
10 

114" 

50' 
40 
30 
20 
10 

113° 

50' 
40 
30 
20 
10 

112° 

50' 
40 
30 
20 
10 

111° 

50' 
40 
30 


20 
10 

110° 

290° 

u 


250° 

70° 

10' 
20 
30 
40 
50 

•71° 

10' 

20 

30 
40 

50 

72° 

10' 
20 
30 
40 

50 

73° 

10' 
20 
30 
40 

50 

74° 

10' 

20 

30 
40 
50 

75° 

10' 

20 
30 
40 
50 

76° 

10' 
20 
30 
40 
50 

77° 
10' 
20 
30 
40 
50 

78° 

10' 
20 
30 
40 
50 

79° 

10' 
20 
30 
40 
50 

80^ 

260° 


43  32.84 
43  35.00 
43  38.33 
43  41.04 
43  43.73 
43  40.40 

43  49.04 
43  51.00 

43  r4.20 
50.84 
59.40 

1.93 

4.44 

0.93 

!>.39 

11.83 

14.25 

10.05 

19.03 
21.38 
23.71 

44  20.02 
44  28.30 
44  30.57 

44  32.82 
44  35.04 
44  37.23 
44  39.40 
44  41.55 
43.08 

45.79 
47.87 
49.93 
51.97 
53.98 
55.97 
57.94 

44  59.89 

45  1.81 


43 
43 
44 

41 
44 
44 
44 
44 
44 

44 
44 
44 


44 

44 
44 
44 
44 
44 
44 

44 


45 
45 
45 

45 
45 
45 
45 
45 
45 

45 
45 
45 
45 
45 
45 

45 
l45 
45 
45 
45 
45 

45 


3.71 
5.59 
7.44 

9.27 
11.08 
12.80 
14.02 

10.30 
18.08 

19.77 

21.44 

23.09 

24.71 

20.31 

27.89 

29.45 

3it.98 

32.49 

33.97 

35.43 

30.87 

38.29 


2.76 

2-73 
2.71 
2.69  i 
2.67 
2.64 

2.O2 

2.60 
2.58 
2.55 

2-53 
2.51 

2.49 
2.46 

2.44 
2.42 
2.40 
2.38 

2-35 

2-33 

2-3' 

2.28 

2.27 

2.25 

2.22 
2.19 
2.17 
215 
2-13 
2.  t  1 

2.0S 
2.06 
2.04 
2.01 
1.99 
1.97 

1-95 
1.92 
1.90 
1. 88 
1,85 
1.S3 

i.Ri 
1.78 
1.7O 
1.74 
1.72 
1.69 

1,67 
,.65 
1.62 
1.60 
1.58 
1.56 

1-53 

1. 48 
1.46 
J. 44 
1.42 


290° 

110° 

50' 
40 
30 
20 
10 

109° 

50' 
40 
30 
20 
10 

108° 

50' 
40 
30 
20 
10 

107° 

50' 
40 
30 
20 
10 

106° 

50' 
40 
30 
20 
10 

105° 

50' 
40 
30 
20 
10 

104 

5(1' 


40 
30 
20 
10 

103° 

50' 
40 
30 
20 
10 

102° 

50' 
40 
30 
20 
10 

101' 

50' 
40 
30 
20 
10 

100' 

280° 

n 


u 

260° 
80° 

10' 

20 

30 
40 
50 

81° 

10' 

20 
30 
40 
50 

82° 

10' 
20 
30 
40 
50 

83° 

10' 
20 
30 
40 
50 

84° 

10' 
20 
30 
40 

50 

85° 

10' 
20 
30 
40 

50 

8G° 

10' 
20 
30 
40 
50 

87' 

111' 
20 
30 

40 
50 

88° 

10' 
20 


45  38.29 
45  39.08 
45  41.05 
45  42.40 
45  43.73 
45  45.03 

45  40.30 

45  47.50 

45  48.79 

45  50.00 

45  51.18 

45  52.34 

45  53.48 
54.59 

55.08 
50.75 
57.79 
58.81 

59.80 
0.78 
1.73 
2.00 
3.50 
4.44 


30 
40 
50 

89° 

10' 
20 
30 
40 
50 

90° 

270° 


45 
45 
45 
45 
45 

45 
40 
40 
40 
40 
40 

40 

40 
40 
40 
40 
40 

40 
40 
40 
40 
40 
40 

40 
41; 
40 
40 
40 
40 

40 
40 
40 
40 
40 
40 

40 
40 
40 
40 
40 
40 


5.30 
0.14 
0.95 
7.74 
8.50 
9.24 

9.90 
10.05 
11.32 
11.97 
12.59 
13.19 

13.70 
14.31 
1  1 .  84 
15.35 
15.83 
10.29 

10.72 
17.13 
17.52 
17.  S!) 
18.23 
18.55 

18.84 
19.11 

19.30 
19.59 
19.79 
19.97 

40  20.12 
40  20.25 
20.30 
20.44 
20.50 
20.53 

20.54 

3 


1-39 
1-37 
1-35 
1-33 
1.30 
1.27 

1.2O 

1-23 
1. 21 
i.iS 
1. 16 
1.14 
I. II 
1.09 
1.07 
1.04 
1.02 
0.99 

0.98 
0-95 
0-93 
0.90 
0.8.S 
0.86 

0.S4 
0.81 
0.79 
0.76 

0.2: 

0.09 

o.r,7i 
0.05 
0.62 
0.60 

°-S7 

°o5 
0-53 
0.51 
0.48 
0.46 
0-43 
0.4T 
°-39 
°-37 
0-34 
0.32 
0.29 


280° 

100° 

50' 
40 
30 
20 
10 

99° 

50' 
40 
30 
20 
10 

98° 

50' 
40 
30 
20 
10 

97° 

50' 
40 
30 
20 
10 

96° 

50' 
40 
30 
20 
10 

95' 

50' 
40 
30 
20 
10 

94° 

511' 
40 
30 
20 
10 

93° 

50' 
40 
30 
20 
10 

92° 


40 
40 
4li 
40 

40 


0.27 

50' 

0.25 

40 

0.23 

30 

0.20 

20 

o.iS 

10 

0.15 

91° 

013 

50' 

0. 1 1 

40 

0.08 

30 

0.06 

20 

0.03 

10 

O.OI 

90° 

270° 

u 

f! 


276 


THE   ORBIT   OF   URANUS. 


Tablk 

XX. 

XXI. 

XXII. 

Aim. 

1 

2 

3 

(6.S.1) 

(6.C..1) 

(6.C.0) 

(6.«.1) 

(6.C.1)  (//.S.2) 

(6.e.2) 

(6.C.0) 

(b.S.l) 

(h.c.V 

(6.M.2I 

(6.C.2) 

// 

// 

tt 

" 

tt 

// 

If 

II 

U 

n 

0 

1.20 

1.12 

0.04 

5.99 

5.42 

0.21 

0.17 

0.0" 

1.58 

1.34 

0.07 

o.n 

10 

1.15 

1.17 

0.03 

6.13 

5.23 

0.19 

0.18 

0. 06 

1.63 

1.14 

0.08 

0.23 

20 

1. 10 

1.21 

0.03 

6.25 

6.04 

r.18 

0.18 

0.(15 

1.67 

0.93 

0.11 

0.29 

30 

1.04 

1.25 

0.(12 

6.34 

4.84 

(,.17 

0.18 

0.04 

1.68 

0.72 

0.15 

0.32 

40 

0.98 

1.28 

0.02 

6.41 

4.62 

0.16 

0.18 

0.03 

1.67 

0.51 

0.21 

0.33 

50 

0.92 

1.31 

0.02 

6.45 

4.40 

0.15 

0.18 

0.03 

1.60 

0.33 

0.27 

0.32 

GO 

0.8(i 

1.33 

0.(12 

6.45 

4.18 

0.16 

0.18 

0.(12 

1.48 

0.18 

0.32 

0.28 

70 

0.79 

1.34 

0.02 

6.40 

3.94 

O.lfi 

0.18 

O.ol 

1.32 

0.09 

0.35 

0.23 

80 

0.72 

1.35 

0.02 

6.32 

3.69 

0.16 

0.17 

0.(!0 

1.13 

0.07 

0.36 

0.18 

90 

O.GIl 

1.35 

0.(13 

6.19 

3.43 

0.17 

0.16 

0.00 

0.94 

0.13 

0.35 

0.13 

100 

0.59 

1.34 

0.03 

G.02 

3.15 

0.17 

0.15 

O.OI 

0.77 

0.25 

0.32 

0.09 

110 

0.52 

1.32 

0.04 

5.81 

2.87 

0.18 

0.14 

0.02 

0.66 

0.41 

0.29 

0.07 

120 

0.4f. 

!.30 

0.04 

5.56 

2.57 

0.19 

0.12 

0.04 

0.62 

•.1.60 

0.25 

0,(1(1 

130 

0.40 

1.27 

0.04 

6.29 

2.26 

0.20 

0.11 

0.06 

0.64 

0.7  V 

0.22 

O.Ofi 

140 

0.34 

1.24 

0.05 

4.99 

1.96 

0.20 

0.10 

0.08 

0.72 

0.91 

0.20 

0.07 

mo 

0.28 

1.20 

0.05 

4  (,'i 

1.66 

0.20 

0.09 

0.09 

0.83 

1.00 

0.18 

0.08 

1(10 

0.23 

1.15 

0. 05 

4.36 

1..S8 

0.20 

0.08 

0.10 

0.95 

1.04 

0.17 

0.09 

no 

0.19 

1.10 

O.OG 

4.03 

1.11 

0.20 

0.08 

O.U 

1.04 

1.04 

0.17 

0.10 

ISO 

0.15 

1.04 

O.OG 

3.69 

0.86 

0.19 

0.07 

0,11 

1.11 

l.OI 

0.16 

0.10 

190 

0.12 

0.98 

0.06 

3.36 

0.65 

0.17 

0.07 

0.11 

1.14 

0.97 

0.14 

0.11 

200 

0.09 

0.92 

O.OG 

3.03 

0.56 

0.16 

0.08 

O.ll 

1.14 

0.93 

0,13 

0.12 

210 

0.07 

0.8*'" 

O.Ofi 

2.71 

0.32 

0.14 

0.08 

0.11 

1.11 

0.91 

0,11 

0.13 

220 

o.or, 

0.79 

O.Ofi 

2.30 

0.20 

0.12 

0.09 

0.10 

1.07 

0.91 

0.10 

0.15 

230 

0.05 

0.72 

O.OG 

2.08 

0.12 

0.1 1 

O.U 

0.10 

1.02 

0.92 

0.10 

0.18 

240 

0.05 

O.GG 

0.07 

1.77 

0.08 

0.09 

0.13 

0.09 

0.98 

0.9C 

0.10 

0.21 

250 

o.on 

0.59 

0.07 

1.48 

0.08 

0.07 

0.15 

0.09 

0.96 

1.00 

0.12 

0.23 

2fiO 

0.08 

0.52 

0.07 

1.21 

0.10 

0.05 

0.17 

0.08 

0. 95 

1.05 

0.14 

0.25 

270 

0.10 

0.4G 

0.08 

1.05 

0.16 

0.03 

0.20 

0.08 

0.94 

1.09 

0.16 

0  26 

2S0 

0.13 

0.40 

0.08 

0.72 

0.26 

0.02 

0.22 

0.07 

0.95 

1.13 

0.19 

0.26 

290 

0.1  fi 

0.34 

0.09 

0.52 

0.38 

0.01 

0.25 

O.Ofi 

0.95 

1.16 

0.22 

0.25 

300 

0.20 

0.28 

0.10 

0.35 

0.54 

0.01 

0.27 

O.Ofi 

0.94 

1.18 

0.23 

0.23 

310 

0.25 

0.23 

0.11 

0.22 

(1.73 

0.01 

0.29 

O.Ofi 

0.93 

1.20 

0.24 

0.21 

320 

0.30 

0.19 

0.12 

0.12 

0.94 

0.02 

0.31 

0.07 

0.91 

1.22 

0,23 

0.19 

330 

0.3rt 

0. 15 

0.13 

0.0(1 

1.18 

0.03 

0,33 

0.08 

0.88 

1.25 

0,22 

0.18 

340 

0.42 

0.12 

0.14 

0.04 

1.44 

0.05 

0.35 

0.09 

0.80 

1.28 

0.20 

0.17 

350 

0.48 

o.no 

0.15 

O.Ofi 

1.72 

0.07 

0.36 

0.09 

0.84 

1.34 

0.18 

0.17 

3r,o 

0.54 

0.(17 

O.lfi 

0.12 

2.01 

0.10 

0.37 

0.10 

0.85 

1.40 

0.16 

0.18 

370 

O.fil 

O.Ul) 

0.17 

0.21 

2.31 

0.13 

0.37 

0.11 

0.88 

1.46 

0.15 

0.19 

380 

0.G8 

0.05 

0.18 

0.34 

2.62 

0.16 

0.37 

0.12 

0.92 

1.51 

0.14 

0.20 

390 

0.74 

0.05 

0.19 

0.50 

2.92 

0.19 

0.36 

0.12 

0.98 

1.54 

0.13 

0.21 

400 

0.81 

O.Ofi 

0.20 

0.70 

3.21 

0.23 

0.35 

0.11 

1.03 

1.54 

0.13 

0.22 

410 

0.88 

0.08 

0.20 

0.92 

3.54 

0.26 

0.31 

0.10 

1.07 

1.50 

0.12 

0.23 

420 

0.94 

0.10 

0.21 

1.17 

3.84 

0.80 

0.33 

0.08 

l.Ofi 

1.44 

0.12 

0.25 

430 

1.00 

0.13 

0.21 

1.45 

4.14 

0.33 

0.31 

0.06 

1.01 

1.37 

0.12 

0.27 

440 

1.06 

O.IG 

0.21 

1.75 

4.42 

0.36 

0.29 

0.04 

0.91 

1.30 

0.12 

0.29 

450 

1.12 

0.20 

0.21 

2.06 

4.70 

0.38 

0.27 

0.03 

0.77 

1.28 

0.12 

0.32 

4  CO 

1.17 

0.25 

0.21 

2  39 

4.96 

0.40 

0.25 

0.02 

O.fil 

1.31 

0.14 

0.35 

470 

1.21 

0.30 

0.20 

2.72 

5.20 

0.41 

0.23 

0,01 

0.47 

1.40 

0.17 

0.38 

4  SO 

1.25 

0.3G 

0.19 

3.06 

5.42 

0.42 

0.21 

0.(11 

0.36 

1.56 

0.21 

0.39 

490 

1.2H 

0.42 

0.18 

3.39 

5.62 

0.42 

0.20 

0.01 

0.32 

1.76 

0.27 

0.39 

500 

1.31 

0.48 

0.17 

3.71 

5.79 

n.42 

0.18 

0.01 

0.36 

1.96 

0.32 

0.37 

510 

1.33 

0.54 

O.lfi 

4.02 

5.92 

0.41 

0.17 

0.01 

0.46 

2.15 

0.37 

0.33 

520 

1.34 

O.Gl 

0.14 

4.31 

6.01 

0.39 

0.16 

0.02 

0.62 

2.21 

0.40 

0.27 

530 

1.35 

0.G8 

0.13 

4.58 

6.06 

0.38 

0.16 

0,02 

0.81 

2.35 

0.40 

5.20 

540 

1.33 

0.74 

O.U 

4.84 

6.08 

0.35 

0.15 

0.03 

1.00 

2.34 

0.38 

0.14 

550 

1.34 

0.81 

0.10 

5.07 

G.Ot 

0.33 

0.15 

0.03 

1.18 

2.2fi 

0.33 

0.08 

5i;o 

1.32 

0.88 

0.09 

5.28 

5.98 

0.30 

0.15 

0.04 

1.31 

2.11 

0.27 

0.05 

570 

1.30 

0.94 

0.07 

5.48 

5.88 

0.28 

0.16 

0.(14 

1.41 

1.94 

0.20 

0.04 

5  so 

1.27 

1.(10 

O.Ofi 

5.67 

5.75 

0.2fi 

0.16 

0.(15 

1.48 

1.74 

0.14 

0.07 

590 

1.24 

l.dCi 

0  05 

6.81 

5.59 

0.23 

0.17 

0.06 

1.53 

1.51 

0,1(1 

0.11 

000 

1.20 

1,12 

0.04 

5.99 

5.42 

0.21 

0.17 

0.(1(1 

l..^S 

1.34 

0.07 

0,17 

THE   ORBIT   OF   URANUS. 


8TT 


T.vm.KXXlH. 


Year.    (6.C.0)  (fc.ti.l),  {b.c.l) 


1300    +0 


fifil— 3.S5'_12.74 


llilO  ;     O.flSl     3.81 


13-20 


0.04, 


.78 


1330  '     0.03      3.74 
1340  I     0.01       3.70 


12.0,0 
12.:)0 
1-2.40 
12.37 


(t.».2)  (6.C.2) 


.0,341—0.07 
0.34  i  0.00 
0.341     0.05 


Year.    (6.c,0)    (ft.s.l)     (b.c.l)  V>-i-'^)\    (l'-'-'-'^) 


1350    +0.00—3.00—12.27 


1300 


0.59;     3.03 


1370  1  0.58 
1380  0.57 
1390        0.50 

1400  +0.55 
0.54  I 
0.53, 
0.52  i 


3.00 
3.57 
3.54 


12.17 
12.07 
11.07 
11.80 


—3.51  —11.70 


1410 
1420 
1430 
1440 


3.48 
3.40 
3.44 


0.51  1  3.43 


1450  +0.5"  —3.42 


11.05 
11.54 
11.43 
11.32 


0.34 
0.34 

-0.34 
0.34 
0.34 
0.34 
0.34 


0.04 
0.03 

'—0.02 
0.00 
0.80 
0.88 
0.87 


1800  +0.15 
1810  ,  0.14 
1820  !  0.13 


1830 


0.12 


1840   0.11 

1850    +0.10 
1800        0.09 


-4.00    —0.00 


1870 


0.08 


_0.35'— 0.80 
0.35  j  0.85 
0.35  0.84 
0.35 1  0.83 
0.35;     0.8- 


1880  :     0.07 


1890 


0.00 


4.08 
4.75 
4.83 
4.92 

-5.00 
5.09 
5.18 
5.27 
5.30 


5.80 
5.01 
5.41 
5.21 


ff 

ft 

—0.44 

—0.40 

0.45 

0.45 

0.45 

0.44 

0.45 

0.43 

0.46 

n  .it\ 

0.42 
n  ii 

—5.00  —0.40   —I 
4.79!     0.47 1 
4.58  1     0.47 


1900  '+0.05!  —5.45 


1010  I     0.04 
1020        0.03 


1030 


0.02 


1400 
1470 
1480 
1490 

1500 
1510 


0.49 
0.48 
0.40 
0.45 

+0.44 
0.43 


-11.21    —0.35 


3.41!  11.10 

3.40 1  10.08 

3.40 1  10.80 

3.40  10.74 

_:?,40  —10.02 
.10       10.50 


1520  0.42 
1530  I  0.41 
1540  i  0.40 


3.41 
3.42 


10.38 
10.20 


3.43   10.14 


1000  '+0.34 
1010  '  0.33 
1C20  i  0.32 
1030  0.32 
1040 

1050 
1 000 
1070 
1080 
1090 

1700 
1710 


1550  +0.39—3.45  —10.01 
1.500  '  0.38:  3.47  0.88 
1570  0.371  3.40  0.75 
1580  0.30!  3.51 
1590   0.351  3.r.3 


0.35 
0.30 
0.30 
0.30 

-0.30 
0.30 
0.30 
0.3- 
0.37 


-0.80 
0.70 
0.78 
0.77 
0.70 


— 0. 


0.7t 
0.73 
(1.72 
0.71 


1040  t  +  0.01 


1950 
1 000 
1070 
1080 
1000 


2000 


0.00 
—0.01 
0.02 
0.04 
0.05 


5.55 
5.04 
5.74 
5.84 

—5.94 
0.04 
6.14 
6.25 
6.30 


0.40 
0.,39 
4.30;  0.48  i   0.38 
4.141  0.48i   0.37 

I 
_3.02  — 0.49  —0.30 

3.09 
3.40 
3.22 
2.97 


0.40 


0.35 


0.50  0.34 
0.50  j  0.33 
0.50  1   0.32 


_')  73—0.51  —0.30 
2.48'  0.51;  0.29 
2.24  0.52!  0.28 
1.99  0.52 1  0.27 
1.73 1  0.521   0.20 


_0.00  —0.47  —1.471— 0.531-0.25 


_n.37  — O.TO 
0.70 


9.01 
0.47 


—3.50  —  0..33 


0.31 


+0.30 
0.20 
0.28 
0.27 
0.20 

+0.25 
0.24 
1720  '  0.23 


3.50 
3.02 
3.05 
3.09 


0.19 
0.05 
8.00 


0.37' 
0.37; 
0.37  I 

0.38 


0.00 
0.08 
0.07 


2010  I  0.07,   0.58 


2020  !  0.0S| 

2030 

2040 


0.00 
0.00  0.80 
0.10  i   0.92 


1.211  0.5:i 


0.04 
0.00 


0.54 
0.54  1 


0.24 
0.23 
0.22 


0.38,  0.541   0.20 


-0.11'  —7.03  L-0.10'-0..55'— 0.19 


2050 

•2000  1  0.12!  7.14  :+o. 


IS   0.55;   0.18 


_0.3S  — 0.00 
0.381  0.05 
0.30!  0.04 


0.30 


0.03 


0.22 
0.21 


1730 
1740 

1750 
1700 
1770 
1780 
1790 


1800    +0.15  — 


3.77 

S.liU      - 

8.44 

3.81 

8.20 

3.85 

8  13 

3.00 

1.00 

—3.05 

—  7.79 

'     4.00 

7.03 

4.00 

7  40 

4.12 

7.20 

4.18 

t.l2 

i  o.\ 

—  0.05 

0.10 

4.32 

0.18 

4.38 

0.17 

4.45 

0.10 
1  n  1  ^ 

4.53 
I  r.n' 

0.30,     0.02 

_0.39'— 0.01 
0.40 1  0.00 
0.40]  0.50 
0.40  0.58 
0.41 


-0.41 
0.41 
0.42 
0.42 
0.42 


0.43 


2070 
2080 
2090 


0.13; 


7.20 


0.15 1   7.37 


0.10 


2100  '—0.17 
2110  !  0.18 
2120  •  0.19 
2130  i  0.21 
2140   O.22I 


7.48 


0.47  0.50 
0.70  0.50 
1.05   0.50 j 


0.17 
0.10 
0.15 


_7  00  +1.35 —0.57' —0.14 
7.72  1  1.05   0.571   "■''* 
7.84 
7.07 
8.09 


2150  —0.23  —8.21 


2100  i  0.241 
2170  '  0.251 
2180   0.27! 


8.33 
8.44 


1.00  0.57; 


0.12 


2.27   0.58   0.10 


2.58 1  0.58 

+2.90—0.58 
3.22i  0.50 
3.541  0.50 
3.87 i  0.50 


0.09 


—0.08 
0.07 
0.00 
0.05 


2100 


0.281 


2200  —0.20  — S 


210 

2220 
2230 


0.301 
0.311 
0.33! 


0.77 

0.43 

0.50 

0.58 

0.43 

0.49 

0.39 

0.44 

0.48 

0.20 

0.44 

0.47 

-  0.00 

—0.44 

—0.40 

2240  ,  0.34 


2250  —0.35  —9 

2200 

2270 

2280 

2290 

2300  —0.42  —10 


0.30 

0.38 

0.30 

0.40 

1 

08 

4.20 

0.00 

0.03 

SO 

+  1.54 

—0.00 

—0.02 

92 

4.88 

0.00 

—0.01 

04 

5.23 

0.01 

0.00 

10 

5  58 

0.01 

+0.01 

.28 

5.94 

0.01 

0.03 

40 

+  0.29 

—0.02 

+0.04 

52 

0.05 

0.02 

0.05 

04 

7.01 

0.03 

0.07 

70 

7.37 

0. 03 

0.08 

.88 

t.74 

0.03 

0.10 

.00 

+8.10 

-0.04 

+  0.11 

(  ( 


!  ii 


in  'i 


TABLE  FOR  FORMING  THE  PRODUCTS  OF  GIVEN  NUMBERS  BY  THE 
SINE  OR  COSINE  OF  A  GIVEN  ANGLE. 


This  tab.  3  is  formed  for  thn  especial  purpose  of  facilitating  the  formation  of  the 
products  («.«.3)  siu  3r,,  (r.c.3)  cos  JJ./,  etc.,  (,.«.!)  sin  g,  (p.cl)  cos  j,  for  entire  degree^ 
tf  q      It  is  so  arranged  that  the  required  products  can  be  taken  out  at  siglit. 
Supposing  the  number  to  be  given  in  seconds  and  decimal  fractions  ot  a  second, 
we  first  seek  the  given  angle  at  the  top  or  bottom  of  the  page,  and  then  enter  one 
of  the  first  nine  lines  of  the  table  with  the  fraction  part  of  the  second,  niterpolatu.g 
for  the  hundredths.   We  then  add  the  result  mentally  to  the  number  corrcspondnig 
to  the  entire  seconds.     The  algebraic  signs  at  the  sides  of  the  angles  are  those  of 
the  sines  or  cosines  corresponding  to  the  angle  and  to  the  column  above  or  below. 
If  the  number  does  not  exceed  3"  we  can  enter  the  table  as  if  it  were  ten  times 
greater,  and  remove  the  decimal  pouit  one  place  to  the  left  ni  the  result. 
For  exanii>le,  to  find  the  value  of 

2l".67  sin  280'  +  2".25  cos  280° 
we  find  the  angle  280°  at  the  bottom  of  a  pair  of  columns,  the  right  hand  one  being 
the  sine  column.  Entering  this  column  with  0.67  as  the  argument,  we  find  O.CUi. 
Entering  with  2.1,  we  find  20.68,  to  which  adding  0.66,  we  have  21".:U  as  the 
sine  product.  Entering  the  other  column  with  22.5,  and  moving  the  decimal  point, 
we  find  0".39  for  the  cosine  product.  Noticing  the  algebraic  signs  on  each  side 
of  280°,  we  find  the  result  to  be  -  21" M  +  0'.39  =  -  20".<J5. 


(  279  ) 


naa 


280 


TAULE   OF    rilODUCTS   OF   SINKS   ALD   COSINES. 


+ 

v+ 

4- 

2"  + 

+ 

3^  + 

+ 

4°+ 

+ 

6°+ 

+  179  — 

+  178  — 

+  177  — 

+  178  — 

+  176  — 

—  181  — 

—  182  — 

—183  — 

—  184  — 

—  185  — 

—359  J- 

—358  + 

-3£ 

gin 

>7  + 

cos 

—  356  + 

—355  + 

sin 

cos 

sin 

cos 

xin 

cos 

sin 

cos 

0.1 

0.00 

0.10 

0.00 

0.10 

0.01 

0.10 

0.01 

0.10 

0.01 

0.10 

0.1 

0.2 

0.00 

0.20 

0.01 

0.20 

0.01 

0.20 

0.01 

0,20 

0.02 

0.20 

0.2 

0.3 

0.01 

0.30 

0.01 

0.30 

0.02 

0.30 

0.02 

0,30 

0.03 

0.30 

0.3 

0.4 

0.01 

0.40 

0.01 

0.40 

0.02 

0.40 

0.03 

0.40 

0.03 

0.40 

0.4 

0.5 

0.01 

0.50 

0.02 

0.50 

0,03 

0,50 

0.03 

0.50 

0.04 

0.50 

0.5 

O.fi 

0.01 

o.(;o 

0.02 

0.(10 

0.03 

0,(10 

0.04 

O.CO 

0.05 

0.(10 

0.6 

0.7 

0.01 

0.70 

0.02 

0.70 

0,04 

0.70 

0.05 

0.70 

0.0(1 

0.70 

0.7 

0.8 

0.01 

o.so 

0.03 

0.80 

0.04 

0.80 

O.Ofi 

0.80 

0.07 

0.80 

0.8 

0.9 

0.02 

0.90 

0.03 

0.00 

0.05 

0.90 

O.Ofi 

0.90 

O.OS 

0.00 

0.9 

1.0 

0.02 

1  00 

0.03 

1.00 

0.05 

1.00 

0.07 

1.00 

0.00 

1.00 

1.0 

2.0 

0.03 

2.00 

0.07 

2.00 

0. 10 

2.00 

0.14 

2.00 

0.17 

1 .  09 

2.0 

3.0 

0.0,) 

3.00 

0.10 

3.00 

O.lfi 

3.00 

0.21 

2.00 

0.2(1 

2.09 

3.0 

4.0 

0.07 

4.00 

0.14 

4.00 

0.21 

3.99 

0.28 

3.99 

0.35 

3.08 

4.0 

6.0 

0.09 

5.00 

0.17 

5.00 

0.20 

4.99 

0..35 

4,09 

0.44 

4.08 

5.0 

6.0 

0.10 

fi.OO 

0.21 

0.00 

0.31 

5,99 

0.42 

5,09 

0.52 

5.08 

6.0 

7.0 

0.12 

7.00 

0.24 

7.00 

0.37 

fi,09 

0.49 

fi.08 

0.61 

6.07 

7.0 

8.0 

0.14 

8.00 

0.28 

8.00 

0.42 

7  09 

0..'>fi 

7.08 

0.70 

7.07 

8.0 

9.0 

O.K! 

9.00 

0.31 

8.09 

0.47 

8.09 

0.03 

8.98 

0.78 

8.07 

9.0 

10.0 

0.17 

10.00 

0,35 

0.90 

0.52 

0.99 

0.70 

0.98 

0.87 

O.Ofi 

10.0 

11.0 

0.1!) 

11.00 

0.38 

10.90 

0.58 

10.08 

0.77 

10.07 

0.96 

1  O.Ofi 

11.0 

12.0 

0.21 

12.00 

0.42 

11.00 

O.fi  3 

11.98 

0.S4 

11.07 

1,05 

11.05 

12.0 

1.3.0 

0.23 

13.00 

0.45 

12.00 

O.fiS 

12.98 

0.01 

12.07 

1,13 

12.05 

13.0 

14.0 

0.24 

14.00 

0.49 

13.99 

0.73 

13.98 

0.98 

13.97 

1.22 

13.05 

14.0 

15.0 

0.2fi 

15.00 

0.52 

14.99 

0.70 

14.98 

1.05 

14.06 

1.31 

14.94 

150 

IG.O 

0.28 

Ifi.OO 

0.5fi 

15.99 

0..S4 

15.98 

1.12 

15.96 

1.30 

15.04 

16.0 

17.0 

0.30 

17.00 

0.50 

Ifi.OO 

O.SO 

lfi.98 

1.10 

16.96 

1.4S 

lfi.04 

17.0 

18.0 

0.31 

IS. 00 

O.C.S 

17.09 

0.04 

17.98 

1.2(; 

17.06 

1.57 

17.03 

18.0 

19.0 

0.33 

10.00 

O.fi  (5 

IS. 99 

0.00 

18.97 

1.33 

18.05 

l.Gfi 

18.93 

19.0 

20.0 

0.3:) 

20.00 

0.70 

10.09 

1.05 

19.07 

1.40 

10.05 

1.74 

19.92 

20.0 

21.0 

0.37 

21.00 

0,73 

20.09 

1.10 

20.07 

1.4fi 

20.05 

1.S3 

20.02 

21.0 

22.0 

0  38 

22,  Ou 

0.77 

21.99 

1.15 

21.07 

1.53 

21.95 

1.02 

2 1 .  02 

22.0 

23.0 

0.40 

23.00 

0,sO 

22.09 

1.20 

22.97 

1 .  (10 

22,04 

2.00 

22.01 

23.0 

24.0 

0.42 

24.00 

0.84 

23.09 

1.2fi 

23.97 

1.(17 

23,94 

2.00 

23.01 

24.0 

25.0 

0.44 

25.00 

0..^7 

24.98 

1.31 

24.97 

1.74 

24  94 

2.18 

24.00 

25.0 

2ti.0 

0.4n 

2(!.00 

»,     ' 

25.98 

1.3fi 

25.90 

l.sl 

25,94 

2.27 

25.00 

2(1.0 

27.0 

0.47 

27.00 

0.04 

2fi.98 

1.41 

2fi.96 

1.S8 

2(1.03 

2.35 

2fi.90 

27.0 

28.0 

0.40 

28.00 

0.08 

27.98 

1.47 

27. 00 

1,05 

27.93 

2.44 

27.  SO 

2S.0 

29.0 

0.51 

20.00 

1.01 

2S.98 

1.52 

28.90 

2.02 

28.93 

2.53 

28.80 

20.0 

30.0 

0.52 

30.00 

1.05 

29.98 

1.57 

29.96 

2.00 

20.03 

2.fil 

20.80 

20.0 

cos 

Bin 

CO.S 

sin 

cos 

sin 

cos 

.sill 

cos 

sin 

+  2- 

71°— 

+  27 

2  — 

+27 

3  — 

+  2- 

14  — 

+  2T 

5  — 

— 2e 

9  — 

—26 

8  — 

— 2£ 

n  — 

—  2( 

56  — 

—26 

5  — 

—  J 

n  + 

—  £ 

(2  + 

—  £ 

3  + 

—  S 

4  + 

—   £ 

•5  + 

+   i 

J9  + 

+  £ 

18  + 

+  £ 

17  + 

+    i 

36  + 

+  £ 

15  + 

' 


J 


TABLE   OP  PRODUCTS  OF   SINES  AND  COSINES. 


281 


J 


+ 

6  + 

+      -7^  + 

+ 

8"+ 

+    r+ 

+    10^ 

+  174  — 

+  173  — 

+  172- 

+  171  — 

^  170  — 

—  186  — 

—  187  — 

—  188  — 

—  189  — 

— 190  — 

—  354  + 

—  358  + 

—352  + 

—  351  + 

—  350  + 

Kill 

CD.i 

ttin 

C(JH 

sin 

cos 

sill 

tus 

Hin 

cos 

0,1 

0.01 

0,10 

0.01 

0,10 

0.01 

0.10 

0.02 

0.10 

0.02 

O.IO 

0,1 

0.2 

0.02 

0,20 

0,02 

0.20 

0.03 

0.20 

0.03 

0.20 

0.03 

0.20 

0.2 

0,3 

0.03 

0..30 

0.01 

0.30 

0.04 

0.30 

0.05 

0.30 

0,05 

0.30 

(1.3 

0,4 

0.04 

0.40 

0.05 

0.40 

0.00 

0.40 

0.06 

0.40 

0.07 

0.39 

0.4 

0.5 

0.05 

0.50 

0.00 

0,50 

0,07 

0,50 

0.08 

0.49 

0.09 

0.49 

0,5 

0.  li 

0.00 

0.00 

0.07 

0,00 

0.08 

0.59 

0.09 

0.59 

0,10 

0.59 

0.0 

0.7 

0,07 

0.70 

0.09 

0,09 

0.10 

0.09 

0.11 

0.09 

0,12 

0.09 

0.7 

O.S 

O.OS 

0.80 

0,10 

0.79 

0.11 

0.79 

0.13 

0.79 

0,14 

0,79 

0.8 

0.9 

O.OD 

0.90 

O.lt 

0.89 

0.13 

0.80 

O.U 

0.«9 

0.10 

0,89 

0.9 

1.0 

O.lO 

0,99 

0.12 

0.99 

0.14 

0.99 

0,10 

0.99 

0.17 

0,98 

1.0 

2.0 

0.21 

1.99 

0.21 

1.99 

0.28 

1.98 

0.31 

1.98 

0  35 

1,97 

2.0 

3.0 

0,31 

2.98 

0.37 

2.98 

0.42 

2.97 

0.47 

2.90 

0.52 

2,95 

3.0 

4.0 

0  42 

3.98 

0,49 

3.97 

O.uO 

3,90 

0.03 

3.95 

0.09 

3,94 

4.0 

5,0 

0.52 

4.97 

0.01 

4.90 

0.70 

4.95 

0.78 

4.94 

0.87 

4,92 

5.0 

fi,0 

0.03 

5.97 

0.73 

5.90 

0.84 

5,94 

0.94 

5.93 

1.04 

5.91 

0,0 

7.0 

0.73 

0.90 

0.85 

0.95 

0.97 

0.93 

1,10 

0.91 

1.22 

0.89 

7.0 

8,0 

0.H4 

7.90 

0.97 

7.94 

1,11 

7.92 

1,25 

7,90 

l..i9 

7.88 

8.0 

9,0 

0.94 

8.95 

l.IO 

8,93 

1.25 

8.91 

1,41 

8.89 

1.50 

8.80 

9.0 

10.0 

1,05 

9.95 

1.22 

9,93 

1.39 

9.90 

1.50 

9.88 

1.74 

9.85 

10.0 

11,0 

1.15 

10,94 

1.34 

10.92 

1.53 

10.89 

1,72 

10.80 

1.91 

10.83 

11.0 

12.0 

1.25 

11.93 

1,40 

11.91 

1.07 

11,88 

1,88 

11.85 

2.08 

11.82 

12,0 

13.0 

1.30 

12,93 

1,58 

12.90 

1.81 

12,87 

2,03 

12.84 

2.20 

12,80 

13.0 

14.0 

1.40 

13.92 

1.71 

13.90 

1.95 

13,80 

2.19 

13.83 

2.43 

13,79 

14.0 

15.0 

1.57 

14.92 

1.83 

14.89 

2.09 

14.85 

2,35 

14.82 

2.00 

14.77 

15,0 

10.0 

1.07 

15.91 

1.95 

15.88 

2.23 

15.84 

2.50 

15.80 

2.78 

15.70 

10. 0 

17. D 

1.7S 

10.91 

2.07      10.87  1 

2.37 

10.83 

2.00 

10.79 

2.95 

10.74 

17.0 

1S,0 

l.ss 

17.90 

2.19 

17.87 

2,51 

17.82 

2,82 

17.78 

3.13 

17.73 

18,0 

19.0 

1.99 

18.90 

2.32 

18.80 

2.04 

18.82 

2.97 

18.7T 

3.30 

18.71 

19,0 

20.0 

2.09 

19.89 

2.44 

10.85 

2.78 

19,81 

3.13 

19.75 

3.47 

19.70 

20.0 

21.0 

2.20 

20.88 

2.50     20.84 

2.92 

20.80 

3.29 

20.74 

3.05 

20.08 

21.0 

22,0 

2.30 

21.88 

2.08     21.84 

3.00 

21.79 

3,44 

21.73 

3.82 

21.07 

22.0 

23.0 

2.40 

22.87 

2.80 

22.83 

3.20 

22,78 

3.00 

22.72 

3.99 

22.05 

23.0 

24.0 

2.51 

23.87 

2.92 

23.82 

3.34 

23,77 

3.75 

23.70 

4.17 

23.04 

24.0 

25.0 

2.01 

24.  SO 

3.05 

24,81 

3.48 

24,70 

3.91 

24.09 

4.31 

24.02 

25.0 

20,0 

2,72 

25.80 

3.17 

25,81 

3,02 

25.75 

4.07 

25.08 

4.51 

25.01 

20.0 

27.0 

2,82 

20.85 

3.29 

20,80 

3.70 

20.74 

4.22 

20.07 

4.09 

20.59 

27,0 

2S.0 

2,93 

27.85 

3,41 

27.79 

3,90 

27.73 

4.38 

27.00 

4.80 

27.57 

28,0 

29.0 

3,03 

28.84 

3.53 

28.78 

4.04 

28.72 

4.54 

28.04 

5.04 

28.56 

29.0 

30.0 

3.14 

29.84 

3.00 

29.78 

4.18 

29.71 

4.09 

29.03 

5.21 

29.54 

30.0 

cos 

sin 

COS 

siti 

cos 

sin 

cos 

sin 

cos 

sin 

+  2' 

76  — 

+  277  — 

+  2' 

IB  — 

+  279  — 

+  2- 

0  — 

—  2( 

84  — 

—  263  — 

—  2( 

32  — 

—  281  — 

—  2( 

30  — 

—    { 

:6  + 

-    97  + 

—    i 

)3  + 

—    89  + 

—  1( 

)0  + 

+    i 

J4  + 

+    83  + 

+    J 

32  + 

+    81  + 

+    i 

JO  + 

3 

a    Aug 

ust.  1873 

. 

282 


TABLE   OF    PRODUCTS   OF   SINKS   AM)   COS  INKS, 


n 


|i!   ; 


+     11°  + 

+     12°  + 

+    13°+ 

+    14°+ 

+    1B°  + 

+  169  — 

+  168   - 

+  167  - 

+  166  — 

+  16B  — 

—  191  — 

—  192  — 

—  193  — 

—  194  — 

—  105  — 

—  349  + 

—  348  + 

—  847  + 

—  346  + 

—  846  + 

sin 

t'08 

«in 

(•(IH 

xln 

cos 

RJn 

cos 

Rin 

cog 

0.1 

0.02 

0.10 

0.02 

0,10 

0.02 

0,10 

0,(12 

0,10 

0.03 

0.10 

0.1 

0.2 

O.lit 

0.20 

O.dt 

0,20 

cot 

0.19 

0,05 

0,19 

0.05 

0.19 

0.2 

0.3 

O.OC, 

0.2'J 

(l,0(> 

0,29 

0.(17 

0.29 

0.07 

0,29 

0.08 

0.29 

0.3 

0.4 

O.OH 

0.39 

0.08 

0,39 

0.09 

0.39 

0,10 

0,39 

0.10 

0.39 

0.4 

O.ft 

O.IO 

0.49 

0.10 

0,(9 

0.11 

0.49 

0.12 

0,49 

0.13 

0.48 

0.5 

O.C) 

0.  II 

0.59 

0.12 

0.59 

0.13 

0,58 

0.15 

0,58 

0,10 

0.58 

O.fi 

0.7 

0.  l:! 

O.CO 

0,15 

0.08 

0. 1 0 

0,08 

0,17 

0.08 

0.18 

0,08 

0.7 

0.8 

o.i:. 

0.79 

0.17 

0.78 

O.IS 

0.78 

0,19 

0,-8 

0,21 

0,77 

0,8 

0.9 

0.17 

0.88 

0.19 

0.88 

0.20 

0.88 

0.22 

0,87 

0,23 

0.87 

0.9 

1,0 

fl.IO 

0.9S 

0,21 

0.98 

0.22 

0.97 

0.24 

0.97 

0.20 

0.97 

1,0 

2.0 

o.;!s 

i.9(; 

0.42 

l.9(; 

0,45 

1,95 

0,48 

1.94 

0.52 

1,93 

2.0 

3.0 

0..M 

2.91 

o.(;2 

2.93 

0,07 

2.92 

0.73 

2,91 

0,78 

2.90 

3.0 

4.0 

0.7(1 

3.93 

0.83 

3.91 

0.90 

3.90 

0.97 

3.88 

1.04 

3.80 

4.0 

f).0 

O.Of) 

4.91 

1,04 

4.89 

1.12 

4,87 

1,21 

4,85 

1.29 

4,83 

5.0 

fi.O 

1.14 

5.89 

1,25 

5.87 

1.35 

5.85 

1,45 

5,s2 

1.55 

5,80 

6.(1 

7.0 

l.:!4 

n.87 

1,4(1 

fi.85 

1.57 

0.82 

1.09 

0.79 

1.81 

0,70 

7,0 

H.O 

i.r>.{ 

7.85 

i,(;r, 

7  83 

1.80 

7,79 

1.94 

7.70 

2.07 

7,73 

8.0 

0.0 

1.72 

8.83 

1.87 

8.80 

2.02 

8.77 

2.18 

8.73 

2.33 

8.09 

9.0 

10.0 

i.ni 

9.82 

2,08 

9.78 

2,25 

9.74 

2,42 

9,70 

2,59 

9.06 

10.0 

11.0 

2.10 

l((.80 

2.29 

10. 7c. 

2^47 

10.72 

2,00 

10.07 

2,s5 

1(1,03 

11.0 

12.0 

') .)() 

11.78 

2,49 

11.74 

2,70 

11,09 

2,90 

11.04 

3,11 

11,59 

12,0 

l.i.O 

2'.4S 

12.7(! 

2,70 

12,72 

2,92 

12,07 

3,14 

12.01 

3.;io 

12.50 

13,0 

14.0 

2.(17 

13.74 

2.91 

13.09 

3.15 

13.G4 

3.39 

13.58 

3.02 

13.52 

14.0 

in.o 

2.Sfi 

14.72 

3,12 

14.07 

3.37 

14,02 

3.03 

14,55 

s.y-H 

14.49 

15.0 

lil.O 

.•J.O,') 

15.71 

3.33 

15.05 

3,00 

15.59 

3.87 

15.52 

4.14 

15.45 

10.0 

17.0 

:i.24 

l(i.(!9 

3.5:i 

10.03 

3,82 

10.50 

4,11 

10.50 

4.10 

10.42 

17.0 

18.0 

.'i.4;! 

17.07 

3,7t 

17.01 

4,(15 

17.54 

4,35 

17.47 

■<,00 

17.39 

18.0 

ly.o 

•J. an 

18. C5 

3.95 

18.58 

4,27 

18.51 

4.00 

18.44 

4,92 

18.35 

19.0 

20.0 

3.S2 

19.03 

4.10 

19.50 

4,50 

19.49 

4.84 

19.41 

5,18 

19.32 

20.0 

21.0 

4.01 

20.01 

4,37 

20.54 

4,72 

20.40 

5.08 

20.38 

5.44 

20.28 

21.0 

22.0 

4.2(1 

2I.(;0 

4,57 

21.52 

4.95 

21.44 

5.32 

21.35 

5.09 

21.25 

22.0 

2:t.0 

4.:i9 

22.58 

4,78 

22.50 

5.17 

22.41 

5,50 

22.32 

5.95 

22.22 

23.0 

24.0 

4.58 

23.50 

4.99 

23.48 

5.40 

23.38 

5.81 

23.20 

0.21 

23.18 

24.0 

25.0 

4.77 

24.54 

5.20 

24.45 

5.02 

24.30 

0.05 

24.20 

0.47 

24.15 

25.0 

2(i.O 

4.!Mi 

25,52 

5.41 

25.43 

5.85 

25, .",3 

0.29 

25.23 

0.73 

25.11 

20.0 

27.0 

5.15 

2(i.50 

5,(;i 

20.41 

0.07 

20,31 

0.53 

20.20 

0.99 

20,08 

27.0 

2.S.0 

5.:!4 

27.49 

5,82 

27.39 

0.3() 

27.28 

0,77 

27.17 

7.25 

27.05 

28.0 

29.0 

5.53 

28.47 

6.03 

28.37 

0.52 

28.20 

7.02 

28.14 

7.51 

28.01 

29.0 

30.0 

5.72 

29.45 

0,24 

29.34 

G.75 

29.23 

7.20 

29.11 

7.70 

2S.98 

30.0 

cos 

sin 

cos 

sill 

cos 

sin 

OIKS 

sin 

t'O.S 

sin 

+  281  — 

+  2£ 

J2  — 

+  2f 

}3  — 

+  2e 

14  — 

+  2f 

J5  — 

—  259  — 

—  2E 

>8  — 

—  257  — 

—  2E 

•8  — 

—  2£ 

>5  — 

—  101  + 

—  IC 

>2  + 

—  103  + 

—  10 

4  + 

—  IC 

)5  + 

+    "79  + 

+    - 

(8  + 

+    77  + 

+    '■ 

16  + 

+    ^ 

'5  + 

TAin 

,K   OF   PRODUCTS   OF    8 

N  !•:  S 

A  Nl) 

C08IT 

s'BS. 

283 

+    10°  + 

+  ir+ 

-t      18°+ 

4    19°+ 

+    20°+ 

+  164  — 

+  163  — 

+  162  — 

+  161  — 

1  160  — 

—  196  — 

—  197  — 

._  198  — 

—  199  — 

—  200  — 

—  344  + 

—  343  + 

—  342  + 

—  341  + 

—  340  + 

Bill 

CHS 

Hin 

cos 

hIii 

C'OH 

KIM       1 

j 

fos 

sill 

cog 

O.l 

0.03 

0.10 

0.03 

0.10 

0.03 

0,10 

0.03  ; 

0.09 

0.03 

0.09 

0.1 

o.a 

0.li)> 

0.1!) 

O.OO 

0.19 

O.OO 

(I  1!) 

0.07 

0. 1  9 

0.07 

0,19 

0.2 

0.3 

o.os 

0.2!) 

0.09   , 

0.29 

0.09 

0.2!) 

0.10     ! 

0  28 

0.10 

0,28 

0.3 

0.4 

0.1 1 

0.38 

0.12 

0.38 

0.12 

0.38 

0.13    , 

0.38 

0.14 

0,38 

0  4 

0.5 

0.14 

0.48 

0.15 

0.48 

0.15 

0.48 

0.10 

0.47 

0,17 

0,47 

0.5 

o.c, 

0.17 

0.58 

0.18 

0.57 

0.19 

057 

0.20 

0.57 

0,21 

0,50 

O.li 

0.7 

0.1!) 

0.07 

0.20 

0.07 

0.22 

0,07 

0,23 

0.06 

0,24 

0,06 

0.7 

0.8 

0.22 

0.77 

0.23 

0.77 

0.25 

0.76 

0,20 

0.76 

0,27 

0,75 

0.8 

0.9 

0.25 

0.87 

0.26 

0.86 

0.28 

0.80 

0.29 

0.85 

0.31 

0,^)0 

0.9 

1.0 

0.28 

0.96 

0.29 

O.OO 

0.31 

0.95 

0.33 

0.95 

0.34 

0,94 

1.0 

L'.O 

0.55 

1.92 

0.58 

1.01 

0.02 

l.!)0 

0,r,5 

1.89 

0,08 

1,88 

2.0 

;i.o 

0.H3 

2.88 

0.S8 

2.87 

0.93 

2.S5 

0.'.' 

2.S4 

1,03 

2.S2 

3.0 

4.0 

1.10 

3.85 

1.17 

3.83 

1.24 

3.80 

1.30 

:;.78 

1,37 

3'70 

4.0 

5.0 

1.38 

4.81 

1.40 

4.78 

1.55 

4.76 

1.03 

4.73 

1.71 

4.70 

5.0 

r,,o 

l.(;5 

5.77 

1.75 

6.74 

1.85 

5.71 

1  95 

5.67 

2.05 

5.04 

6.0 

V.o 

1.113 

0.73 

2.05 

6.09 

2.10 

0.06 

,.28 

f..02 

2.3  9 

0.58 

7.0 

8.0 

2.21 

7.09 

2.34 

7.05 

2.47 

7.01 

2.00 

7.56 

2.74 

7.52 

8.0 

n.,; 

2.48 

8.65 

2.03 

8.01 

2.78 

8.56 

2.93 

8.51 

3.08 

8.40 

9.0 

10.0 

2.7(5 

0.01 

2.92 

9.56 

3.09 

0.51 

3.0') 

9.40 

3.42 

9.40 

lO.O 

11.0 

3.03 

10.57 

3.22 

10.52 

3.40 

10.46 

3,5X 

10.40 

3.76 

10.34 

n.o 

12.0 

3.31 

11.54 

3.51 

11.48 

3,71 

11.41 

3,91 

11.35 

4.10 

11.28 

12.0 

13.0 

3.58 

12.50 

3.80 

12.43 

4.02 

12.36 

4,23 

1::.29 

4.45 

12.22 

13.0 

14.0 

3.80 

13.46 

4.09 

13.39 

4.33 

13.31 

4.50 

13.24 

4.7!) 

13.10 

14.0 

15.0 

4.13 

14.42 

4.39 

14.34 

4.04 

14.27 

4.88 

14. IS 

5.13 

14.10 

15.0 

ICO 

4.41 

15.38 

4.08 

15.30 

4.!I4 

15.22 

5.21 

15.13 

5.47 

1  J. 04 

10.0 

n.o 

4.  CO 

10.34 

4.97 

10.26 

5.25 

10.17 

5.53 

16.07 

5.81 

i5.!»7 

17.0 

is.O 

4.',)li 

17.30 

5.26 

17.21 

5.56 

17.12 

5.80 

17.02 

0.10 

10.91 

18,0 

10.0 

5.24 

18.26 

5.56 

18.17 

1 

5.87 

18.07 

0.19 

17.96 

6.50 

17.85 

19.0 

20.0 

5.51 

19.23 

5. 85 

19.13 

0.18 

19.00 

0.51 

IS. 91 

O.fvl 

18.79 

20.0 

21.0 

5. 79 

1  20.19 

0.14 

20.08 

0.4  9 

1  1!1.'.I7 

6.84 

19.  MO 

T.18 

;   19.73 

21.0 

22.0 

fi.Oll 

21.15 

0.43 

21.04 

0.80 

2().!t2 

7.10 

20.80 

7.52 

20.07 

22.0 

23.0 

0.34 

22.11 

0.72 

!  22.00 

7.11  i  'Jl.87 

7.49 

21.75 

7.87 

21.01 

23.0 

24.0 

G.02 

23.07 

7.02 

22.05 

1 

7.42 

22.83 

7.81 

22.60 

8.21 

22.55 

1 

24.0 

25.0 

O.SO 

24.03 

7.31 

] 

23.91 

7.73 

23.78 

8.14 

2  ".04 

8.55 

23.49 

25  0 

20. 0 

7.17 

24.99 

7.00 

24.80 

S.(K! 

2t.73 

8.40 

'J4.58 

8.8!) 

24.43 

20.0 

27.0 

7.44 

25.05 

7.89 

25.82 

8.34 

25.08 

S.79 

25.53 

9.23 

25.37 

27.0 

2S.0 

7.72 

'  20.92 

8,19     20.78 

8.05 

20.03 

9.12 

i   20.47 

9.58 

20.31 

28,0 

2'J.O 

T.it'J 

27.88 

8.48 

27.73 

8.90 

27.58 

j 

9.44 

27.42 

9.92 

27.25 

29.0 

30.0 

8.27 

28.84 

8.77 

28.69 

9.27 

,  28.53 

9  7T 

28.37 

10.26 

28.19 

30.0 

con 

sin 

cos 

sin 

co.s 

sia 

cus 

BlU 

C08 

I      sin 

+  286  — 

+  287  — 

+  288  — 

+  2 

89  — 

+  290  — 

—  254  — 

—  253  — 

—  252  -  - 

—  2 

51  — 

—  250  — 

—  106  + 

—  107    f 

—  108  + 

—  1 

09  + 

—  110  + 

+   "74  + 

+    73  + 

+  ^2  + 

+ 

71  + 

+    70  + 

■dB 


284 


TABLE   OF   PKODUCTS   OF    SINES   AND   COSINES, 


+ 

21  + 

+    22°  + 

+    23'  + 

+    24°+ 

+    25°+ 

■i  159  — 

+  158  — 

+  157  — 

+  156  — 

+  155  — 

—  201  — 

—202  — 

—  203  — 

—  204  — 

—  205  — 

—  839  + 

—  338  + 

—  337  + 

—  336  + 

—  335  + 

sin 

cos 

Hin 

cos 

sin 

cos 

sin 

COS 

sin 

COS 

0.1 

0.04 

0.09 

0.04 

0.09 

0.04 

0.09 

0.04 

0.09 

0.04 

0.09 

0.1 

0.2 

0.07 

0.19 

0.07 

0.19 

0.08 

0.18 

0.08 

0.18 

0.08 

0.18 

0.2 

0.3 

O.ll 

0.28 

0.11 

0.28 

0.12 

0.28 

0.12 

0.27 

0.13 

0.27 

0.3 

0.4 

0.14 

0.31 

0.15 

0.37 

0.10 

0.37 

0.16 

0.37 

0.17 

0.36 

0.4 

0.5 

O.IS 

0.4t 

0.19 

0.46 

0.20 

0.46 

0.20 

0.46 

0.21 

n.45 

0.5 

O.G 

0.22 

0.56 

0.22 

0.56 

0.23 

0.55 

0.24 

0.55 

0.25 

0.54 

0.6 

0.7 

0.25 

0.05 

0.20 

0.05 

0.27 

0.04 

0.28 

0.04 

0.30 

0.63 

0.7 

O.S 

0.2!) 

0.75 

0.30 

0.74 

0.31 

0.74 

0.33 

0.73 

0.34 

0.73 

0.8 

0.9 

0.32 

0.84 

0.34 

0.83 

0.35 

0.83 

0.37 

0.82 

0.38 

0.82 

0.9 

1.0 

o..3r. 

0.93 

0..37 

0.93 

0..S9 

0.92 

0.41 

0.91 

0.42 

0.91 

1.0 

2.0 

0.72 

1.87 

0.75 

1.85 

0.78 

1.S4 

0.81 

1.83 

0.85 

1.81 

2.0 

3.0 

l.OS 

2.80 

1.12 

2.78 

1.17 

2.76 

1.22 

2.74 

1.27 

2.72 

3.0 

4.0 

1.43 

3.73 

1.50 

3.71 

1.56 

3.08 

1.03 

3.65 

1.69 

3.63 

4.0 

5.0 

1.70 

4.67 

1.S7 

4.04 

1.95 

4.60 

2.03 

4.57 

2.11 

4.63 

5.0 

().0 

2.15 

5.60 

2.25 

5.56 

2.34 

5.52 

2.44 

5.48 

2.54 

5.44 

0.0 

7.0 

2.51 

0.54 

2.1  ' 

0.49 

2.74 

6.44 

2.85 

6.39 

2.96 

6.34 

7.0 

8.0 

2.H7 

7.47 

3.00 

7.42 

3.13 

7.36 

3.25 

7.31 

3.38 

7.25 

8.0 

U.O 

3.23 

8.40 

3. 37 

8.34 

3.52 

8.28 

3.06 

8.22 

3.80 

8.16 

9.0 

10.0 

3.58 

9.34 

3.75 

9.27 

3.91 

9.21 

4.07 

9.14 

4.23 

9.00 

10.0 

U.O 

3.94 

10.27 

4.12 

10.20 

4.30 

10.13 

4.47 

10.05 

4.05 

9.97 

11.0 

12.0 

4.30 

11.20 

4.50 

11.13 

4.09     11.05 

4.88 

10.96 

6.07 

10.88 

12.0 

13.0 

4.(ii> 

12.14 

4.87 

12.05 

5.08 

11.97 

5.29 

11.88 

5.49 

11.78 

13.0 

14.0 

5.02 

13.07 

5.24 

12.98 

5.47 

12.89 

5.69 

12.79 

6.92 

12.69 

14.0 

15.0 

5.3S 

14.00 

5.02 

13.91 

5.80 

13.81 

6.16 

13.70 

6.34 

13.59 

15.0 

Ki.O 

5.73 

14.94 

5.99 

14.83 

0.2;. 

14.73 

6.51 

14.02 

6.76 

14.50 

10.0 

n.o 

(l.O!) 

15.S7 

0.37 

15.70 

6.04 

15.65 

0.91 

15.53 

7.18 

15.41 

17.0 

IH.O 

fi.45 

10.80 

0.74 

10.09 

7.03 

10.57 

7.32 

10.44 

7.61 

10.31 

18.0 

I'J.O 

6.x  I 

17.74 

7.12 

17.02 

7.42 

17.49 

7.73 

17.36 

8.03 

17.22 

19.0 

20.0 

7.17 

18.07 

7.19 

18.54 

7.81 

18.41 

8.13 

18.27 

8.45 

18.13 

20.0 

21.0 

7.53 

19.01 

7.S7 

19.47 

8.21 

19.33 

8.54 

19.18 

8.S7 

19.03 

21.0 

22.0 

7.S.S 

20.54 

8.24 

20.40 

8. CO 

20.25 

8.95 

20.10 

9.30 

19.94 

22.0 

23.0 

8.24 

21.47 

8.li2 

2 '..33 

8.99 

21.17 

9.35 

21.(»1 

9.72 

20.85 

23.0 

24.0 

8.00 

22.41 

8.99 

22.25 

9.38 

22.09 

9.70 

21.93 

10.14 

21.75 

24.0 

25.0 

8.  or, 

23.34 

9.37 

23.18 

9.77 

23.01 

10.!7 

22.84 

10.57 

22.66 

25.0 

2(1.0 

9.32 

24.27 

9.74 

24.11 

10.10 

23.93 

10.58 

23  75 

10.99 

23.58 

20.0 

27.0 

O.C.S 

25.21 

10.11 

25.03 

10.55 

24.85 

10.98 

24.07 

11.41 

24.47 

27.0 

2S.0 

10.03 

20.14 

10.49 

25.96 

10.94 

25.77 

11.39 

25.58 

11.83 

25.38 

28.0 

2i».0 

10.39 

27.07 

10.8(1 

26.89 

11.33 

20.69 

11.80 

20.49 

12.26 

26.28 

29.0 

30.0 

10.75 

)  28.01 

11.24 

27.82 

11.73 

27.62 

12.20 

27.41 

12.68 

27.19 

30.0 

cos 

.sin 

co.s 

sin 

cos 

sin 

cos 

sin 

COS 

sin 
6  — 

+  2 

91  — 

+  2! 

?2  — 

+  293  — 

+  29 

4  — 

+  2£ 

—  2- 

49  — 

-2' 

18  — 

—  OAl  — 

—  24 

[B  — 

—  24 

t5  - 

—  1 

11  + 

—  1 

12  + 

—  113  + 

—  11 

4  + 

—  11 

6  + 

+ 

89  + 

+    i 

J8  + 

+    67  + 

+  e 

8  + 

-r    € 

15  + 

TABLE   OP   PRODUCTS ( 

OF    SINES 

AND 

COSINES. 

285 

+    26"+ 

+    27"  + 

+    28°+ 

+    29"  + 

+    30'+ 

+  154  — 

+  153  — 

+  152  — 

+  151  — 

+  150  — 

—  206  — 

—  207  — 

—  208  — 

—  209  — 

—  210  — 

—  334  + 

—  333  + 

—  332  + 

—  331  + 

—  330  + 

sin 

cos 

sin 

cos 

sin 

cos 

sin 

cos 

sin 

cos 

0.1 

0.04 

0.00 

0.05 

0,00 

0.05 

0.00 

0.05 

0.09 

0.05 

0.09 

0.1 

0.2 

0.09  : 

0.18 

0.00 

0.18 

0.00 

0.18 

0.10 

0.17 

0.10 

0.17 

0.2 

0.:^ 

0.13  : 

0.27 

0.14 

0.27 

0.14 

0.26 

0.15 

0.26 

0.15 

0.26 

0.3 

0.4 

O.IS 

0.36 

0.18 

0.36 

0.10 

0.35 

0.10 

0.35 

0.20 

0.35 

0.4 

0.5 

0.22 

0.45 

0.23 

0,45 

0.23 

0.44 

i 

0.24    ! 

0.44 

0.25 

0.43 

0.5 

O.fi 

0.26  1 

0.54 

0,27 

0,53 

0.28 

0.53 

0.20 

0.52 

0.30 

0.52 

0.6 

0.7 

0.!J1 

0.63 

0.32 

0.62 

0.33 

0.62 

0.34 

0.61 

0.35 

O.Cl 

0.7 

0.8 

0.35 

0.72 

0.36 

0.7i 

0.38 

0.71 

0.30 

0.70 

0.40 

0.69 

0.8 

0.9 

0.39 

O.sl 

0.41 

0.80 

0.42 

0.79 

0.44 

0.70 

0.45 

0.78 

0.0 

1.0 

0.44 

0.00 

0.45 

0.80 

0.47 

0.88 

0.48 

0.87 

0.50 

0.87 

1.0 

2.0 

O.SH 

1.80 

0.01 

1.78 

o.ot 

1.77 

0.07 

1.75 

1.00 

1.73 

2.0 

y.o 

1.32 

2.70 

1.36 

2.67 

1.41 

2.65 

1.45 

2.62 

1.50 

2.60 

3.0 

4.0 

1.75 

3.60 

1.82 

3.56 

1.88 

3.53 

1.04 

3.50 

2.00 

3.46 

4.0 

5.0 

2.10 

4.40 

2.27 

4.46 

2.35 

4.41 

2.42 

4.37 

2.50 

4.. 33 

50 

c.o 

2.(53 

5.30 

2.72 

5.35 

2..S2 

5.. 30 

2.01 

5.25 

3.00 

5.20 

6.0 

t.o 

3.07 

6.20 

3.18 

6.24 

3.20 

6.18 

3.30 

6.12 

3.50 

6.06 

7.0 

s.u 

3.51 

7.10 

3  63 

7.13 

3.711 

7.06 

3.88 

7.00 

4.00 

6.03 

8.0 

9.0 

3.0,". 

8.09 

4.00 

8.02 

4.23 

7.05 

4.36 

7.87 

4.50 

7.70 

0.0 

10.0 

4.3^^ 

8.00 

4.51 

8.01 

4.60 

8.83 

4.85 

8.75 

5.00 

8.66 

10.0 

11.0 

4.S2 

0.80 

4.00 

0.80 

5.16 

0.71 

5.33 

9.62 

5.50 

0.53 

11.0 

12.0 

5.2(1 

10.79 

5.45 

10.69 

5.63 

10.60 

5.82 

10.50 

6.00 

10.30 

12.0 

13.0 

5.70 

• 

11.68 

5.00 

11. 5S 

6.10 

11.48 

6.30 

11.37 

6.50 

11.26 

13.0 

14.0 

6.14 

12.58 

6.36 

12.47 

6.57 

12.36 

C.79 

12.24 

7.00 

12.12 

14.0 

l.'i.O 

fi.58 

13.48 

fi.Sl 

13.37 

7.04 

13.24 

7.27 

13.12 

7.50 

12.09 

15.0 

)(>.0 

7.01 

14.38 

7.26 

14.26 

7.51 

14.13 

7.76 

13.00 

8.00 

13.86 

16.0 

n.o 

7.45 

15.28 

7.72 

15.15 

7.08 

15.01 

8.24 

14.87 

8.50 

14.72 

17.0 

IH.O 

7.  SO 

16.18 

8.17 

16.04 

8.45 

15.89 

8.73 

1.5.74 

9.00 

15.50 

18.0 

19.0 

8.33 

17.08 

8.63 

16.03 

8.92 

16.78 

9.21 

16.62 

9.50 

16.45 

10.0 

20.0 

8.77 

17.08 

O.OS 

17.82 

0.30 

17.66 

0.70 

17.40 

10.00 

17.32 

20.0 

21.0 

0.21 

:  18.87 

0,53 

18.71 

0..'S(1 

is.,'^4 

10.18 

18.37 

10.50 

18.10 

21.0 

22.0 

0.(14 

10.77 

0.00 

10.60 

10.33 

10.42 

10.67 

10.24 

11.00 

10.05 

22.0 

2:1.0 

10.  OS 

20.67 

10.44  ■  20.40 

10.80 

20.31 

11.15 

20.12 

11.50 

10.02 

23.0 

24.0 

10.52 

21.57 

10.00 

21.38 

11.27 

;  21.10 

11.64 

20.09 

12.00 

20.78 

24.0 

25.0 

10.06 

22.47 

11.35 

22. 2S 

11.74 

22.07 

12.12 

21.87 

12.50 

21.65 

25  0 

2li.0 

11.40 

23.37 

11.80     23,17 

12.21 

22.06 

12.61 

\  22.74 

13.00 

22.52 

26.0 

27.0 

11. S4 

24.27 

12.26     24.06 

12.68 

23.84 

13.00 

23.61 

13.50 

23.38 

27,0 

2H.0 

12.27 

25,17 

12.71     24.05 

13.15 

24,72 

13.57 

24.40 

14.00 

24.25 

2S.0 

2'.).0 

12.71 

26.07 

13.17 

25.84 

13.61 

25,61 

1 

14.06 

25.36 

14.50 

25.11 

20.0 

30.0 

13.15 

26.06 

13.62 

i 

'  26.73 

14.08 

!  26.49 

14. .M 

26.24 

15.00 

25.08 

30.0 

CO..! 

.sin 
96  — 

cos 

sin 

cos 

sin 

cos 

sin 

cos 

sin 

+  2 

+  297  — 

+  298  — 

+  299  — 

+  300  — 

—3 

44  — 

—  243  — 

—  242  — 

—  241  — 

—  240  — 

—  ] 

Lie  + 

—  117  + 

—  118  + 

-119  + 

— 120  + 

+ 

64  + 

+    63  + 

+    62  + 

+    61  + 

+    60  + 

286 


TABLE    OF   niODUCTS   OF   SIXES   AND   COSINES. 


t 


1 1 1 


0.1 
0.2 
0.3 
0.4 

0.5 
0.6 
0.7 
0.8 
0.9 

1.0 
2.0 
8.0 
4.0 

5.0 
(i.O 
7.0 
«.0 
!».0 

10.0 
11.0 
12.0 
l.f.O 
14.0 

15.0 

in.o 

17.0 
IH.O 
19.0 

20.0 
21.0 
22.0 
2:t.O 
24.0 

25.0 
2tl.O 
27  0 
2S.0 
29.0 

30.0 


+    31°  + 

+i4r  — 

—  211  — 
—329  + 


sm 

0.05 
0.10 
0.15 
0.21 

0.20 
0.31 
0.30 
0.41 
0.40 

0.52 
1.03 
1.55 
2.01! 

2.5S 
3.09 
3.01 
4.12 
4. 04 

5.15 
5.(;7 
fi.lS 
0.70 
7.21 

7.73 
8.24 
8.70 
9.27 
9.79 

10.30  I 

li).X2 

11.33 

il.85 

12.30 

12.88 
13.39 
13.91 
14.42 
14.94 

15.45 

(■C).S 


+  32°+ 
+148  — 
—212  — 
—328  + 


cos 

0.09 
0.17 
0.26 
0.34 

0.43 
0.51 
0.00 
0.09 
0.77 

O.Sfi 
1.71 
2.57 
3.43 

4.29 
5.14 
6.00 
0.86 
7.71 


17.14 
18.00 

18.86 
19.71 
20.57 

21.43 
22.29 
23.14 
24.00 

24.86 

25.71 
sin 


+  301°— 
—239  — 
—  121  + 

+  59  + 


sm 

0.05 
0.11 
0.16 
0.21 

0.26 
0.32 
0.37 
0.42 
0.48 

0.53 
1.06 
1.59 
2.12 

2.65 
3.18 
3.71 
4.24 
4.77 

5.30 
5.  S3 
6.36 


89 
42 

95 
18 
01 


I 


0.08 
0.17 
0.25 
0.34 

0.42 
0.51 
0.59 
0.68 
0.76 

0.85 
1.70 
2.54 
3.39 

4.24 
5.09 
5.94 

6.78 
7.63 

8.48 

9.33 

10.18 

11.02 

11.87 

12.72 
13.57 
14.42 
15.26 


9.54 
10.07     16.11 


10.60 
11.13 
11.66 
12.19 
12.72 

13.25 
13.78 

I4.:!i 

14.84 
15.37 


16.96 
17.81 
18  lu; 

19.51 
20.35 

21.20 
22.05 
22.90 
23.75 
24.59 


15.90     25.44 
co.s        sin 


+  302  — 
—238  — 
—  122  + 

+  58  + 


+  33°+ 
+147  — 
—213  — 
—327  + 


sill 

0.05 
0.11 
0.16 
0.22 

0.27 
0.33 
0.38 
0.44 
0.49 


CU9 

I  0.08 
0.17 
0.25 
0.34 

0.42 
0.50 
0.59 
0.67 
0.75 

0.84 
1.68 
2.52 
3.35 

4.19 
5.03 
5.87 
6.71 
7.55 

H.39 

9.23 

10.06 

10.90 

11.74 

12.58 
13.42 
14.26 
15.10 
10.35  :  15.93 


5.45 

5.99 
6.54 
7.08 
7.62 

8.17 
H.71 
9  26 
9.80 


10.89 
11.44 
11.98 
12.53 
13.07 

13.62 
14.16 
14.71 
15.25 
15.79 


16.77 
17.61 
1.S.45 
19.29 
20.13 

20.97 
21.81 
22.64 
23.48 
24.32 


16.31     25.16 
C(is        sill 

+303  — 
—237  — 
—  123  + 

+   57  + 


+  34°+ 
+  148  — 
—214  — 
—  326  + 


0.06 
0.11 
0.17 
0.22 

0.28 
0.34 
0.39 
0.45 
0.50 

0.56 
1.12 
1.68 
2.24 

2.80 
3.36 
3.91 
4.47 
5  03 


5. 

6. 
6 

7. 
7. 

8. 

8. 

9. 
10. 
10 


59 
15 
71 

,27 
83 

39 
95 
51 

07 
62 


11.18 
11.74 
12.30 
12.86 
13.42    '   19.90 


cos 

0.08 
0.17 
0  25 
0.33 

0.41 
0.50 
0.58 
0.66 
0.75 

0.83 
1 .  66 
2.49 
3.32 

4.15 

4.97 
5.. 80 
6.63 
7.46 

8,29 

9. 1 2 

9.95 

10.78 

11.61 

12.44 
13.26 
1(.09 
14.92 

15.75 

16.58 
17.41 
18.24 
19.07 


13.98 
14.54 
15.10 
15.  (',6 
16.22 


20.73 
21.55 
22.38 
23.21 
24.04 


16.78    1  24.87 
COS  sin 


+  304  — 

—  236  — 

—  124  + 

+    58  + 


+   35°+ 
+  145  — 

—  215  — 

—  325  + 


0.06 
0.11 
0.17 
0.23 

0.29 
0.34 
0.40 
0.46 
0.52 

0.57 
1.15 
1.72 
2.29 


cos 

0.08 
0. 1 6 
0.25 
0.33 

0.41 
0.49 
0..57 
0.66 
0.74 

0.82 
1.64 
2.46 
3.28 


2.87  '■  4.10 

3.44  i  1.91 

4.02  i  5.73 

4.59  !  6.55 

6.16  '  7.37 


5.74 
6.31 

6.S8 
7.46 
8.03 

8.60 

9.18 

9.75 

10.32 

10.90 

11.47 
12.05 
12.62 
13.19 
13.77 


8.19 

9.01 

9.  S3 

10.65 

11.47 

12.29 
13.11 
13.93 
14.74 
15.56 

16.38 
17.20 
is. 02 
1S.,S4 
19.66 


14.34  I  20.48 

14.91  i  21.30 

15.49  22.12 

16.06  22.94 

16.63  23.76 


17.21 


24.57 
sin 


+  305  — 
—235  — 
—  125  + 

+   55  + 


0.1 

0.2 
0.3 
0.4 

0.5 
0.6 
0.7 
0.8 
0.9 

1.0 

2.0 
3.0 
4.0 

5.0 
6.0 
7.0 
8.0 
9.0 

10.0 
11.0 
12.0 
13.0 
14.0 

15.0 
16.0 
17.0 
IS.O 
19.0 

20.0 
21.0 
22.0 
23.0 
24.0 

25.0 
26.0 
27.0 
28. 0 
29.0 

30.0 


TABLE   OF   PK0DUCT8   OF   SINES  AND    COSINES. 


287 


+    36°+ 

+    37°+ 

+    38°+ 

+    39°+ 

+    40°+ 

+  144  — 

+  143  — 

+ 142  — 

+  141  — 

+  140  — 

—  216  — 

—  217   - 

—  218  — 

—  219  — 

—  220  — 

—  324  + 

—  323  + 

—  322  + 

—  321  + 

—  320  + 

sin    j 

cos 

sin 

cos 

sin 

cos 

sin 

cos 

sin 

co.s 

0.1 

0.00 

0.08 

0.00 

0,08 

0.00 

0.08 

0.00 

0,08 

0.00 

0.08 

0.1 

0.2 

0.12 

0.10 

0.12  1 

0,10 

0  12 

0.10 

0.13   ! 

0.10 

0.13 

0.15 

0.2 

0.3 

0.18 

0.24 

0.18  ! 

0,24 

0.18 

0.24 

0.19 

0.23 

0.19 

0.23 

0.3 

0.4 

0.24 

0.32 

0.24 

0.32 

0.25 

0.32 

0.25 

0.31 

0.20 

0.31 

0.4 

0.5 

0.20 

0.40 

0.30 

0.40 

0,31 

0.39 

0.31 

0.30 

0.32 

0.38 

0.5 

0.0 

o.^;'^ ; 

0.49 

0.30 

0.48 

0.37 

0.47 

0.38 

0.47 

0.39 

0.40 

0.0 

0.7 

0.41  > 

0,57 

.0.42 

0.50 

0.43  i 

0  55 

0.44    ■ 

0.54 

0.45 

0.54 

0.7 

0.8 

0.47 

0.05 

0.48  i 

0,04 

0.40 

0.03 

0.50 

0,02 

0,51 

0.01 

0.8 

O.U 

o.5:j  . 

0.73 

0.54 

0.72 

0.55 

0.71 

0.57 

0.70 

0.58 

0.09 

0.9 

1.0 

0.59  i 

0.81 

0.00 

0.80 

0.02 

0.79 

0.03 

0.78 

0.04 

0.77 

1.0 

2.0 

1.18 : 

1.02 

1.20 

1.00 

1.23 

1.58 

1.20  ; 

1.55 

1.29 

1.53 

2.0 

;{.o 

1.70 

2.43 

1.81 

2.40 

1.85 

2.30 

1,80 

2,33 

1.93 

2.30 

3.0 

4.0 

2.^5 

3.24 

2.41 

3.19 

2.40 

3.15 

2,52 

3.11 

2.57 

3.06 

4.0 

5.0 

2.04 

4.05 

3.01 

3.09 

3.08 

3.04 

3.15 

3,80 

3.21 

3.83 

5.0 

0.0 

;{.:):! 

4.85 

3.01 

4.79 

3.(;o 

4,73 

3.78 

4.00 

3.86 

4.00 

6.0 

7.0 

4.11 

5.00 

4.21 

5.59 

4.31 

5,52 

4.41 

5.44 

4.50 

5.30 

7.0 

8.0 

4.70 

0.47 

4.81 

0,39 

4.93 

0,30 

5.03 

0.22 

5.14 

6.13 

8.0 

9.0 

5.29 

7.28 

5.42 

7.10 

5.54 

7,09 

5.00 

0.90 

5.79 

0.89 

9.0 

10.0 

5. 88 

8.00 

0.02 

7.99 

0.10 

7.88 

6.29 

7.77 

6.43 

7.00 

10.0 

11.0 

0.47 

8.90 

0.02 

8.78 

0.77 

8.07 

0,92 

8.55 

7.07 

8.43 

11.0 

li'.O 

7.05 

9.71 

7.22 

9.58 

7.39 

9.40 

7.55 

9.33 

7.71 

0.10 

12.0 

l.i.O 

7.04 

10.52 

7.82 

1(1.38 

8.00 

10.24 

8.18 

10.10 

8.30 

9,90 

13.0 

14.0 

8.23 

11.33 

8.43 

11.18 

8.02 

11.03 

8.81 

10.88 

9.00 

10,72 

14.0 

l.'i.O 

8.82 

12.14 

9.03 

11,98 

0.23 

11.82 

0.44 

11.00 

9.04 

11.49 

15.0 

10.(1 

9.40 

12.94 

9.03 

12.78 

9.85 

12.01 

10.07 

12.43 

10.28 

12.20 

10.0 

n.o 

9.09 

13.75 

10,23 

13.58 

10,47 

13.40 

10.70 

13.21 

10.93 

13.02 

17.0 

IH.O 

10. 5S 

14.50 

10.83 

14.38 

11.08 

14.18 

11.33 

13.99 

11.57 

13.79 

18.0 

19.0 

11.17 

15.37 

11.43 

15.17 

11.70 

14.97 

11.90 

14.77 

12.21 

14.55 

19.0 

20.0 

11.70 

10.18 

12.04 

15.07 

12,31 

15.70 

12.50 

15.54 

12.80 

15.32 

20.0 

21.0 

12.34 

10.99 

12.04 

10.77 

12.93 

1(;.55 

13.22 

10.32 

13.50 

10.09 

21.0 

22.0 

12.0.{ 

17.80 

13,24 

17.57 

13.54 

17.34 

13.85 

17.10 

14.14 

10.85 

22.0 

2.'!.0 

1.!.52 

18.(il 

1.3.84 

'  18.37 

14.10 

18.12 

14.47 

17.87 

14,78 

17.02 

23.0 

24.0 

14.11 

19.42 

14.44 

10.17 

14.78 

;  18.91 

15.10 

18.(i5 

15.43 

18.39 

24.0 

2:").0 

14.09 

20.23 

15.05 

10,97 

15.30 

19.70 

15.73 

19.43 

10.07 

19.15 

25.0 

21). 0 

15.28 

21.03 

15.05 

2(1.70 

IC.dl 

20.49 

10.30 

20.21 

10.11 

19.92 

20.0 

27.0 

15.87 

21.84 

10.25 

21.50 

10.02 

21. 2« 

10.09 

i  20.08 

17.30 

20.08 

27.0 

2^.0 

1(').40 

22.05 

10.85 

22,30 

17.24 

22,(10 

17.02 

21.70 

18.00 

21.45 

28.0 

29.0 

17.05 

23.40 

17.45 

23.10 

17.85 

22,85 

\ 

18.25 

22.54 

18.04 

22.22 

20.0 

30.0 

17.03 

24.27 

18.05 

23.90 

18.47 

23.04 

18.88 

23.31 

19.28 

22.98 

30.0 

cog 

sin 

cos 

sin 

(,'os 

sin 

cos 

sin 

cos 

1      sin 

+  308  — 

+  307  — 

+  308  — 

+  309  — 

+  310  — 

—  234  — 

—  -233  — 

—  232  — 

—  231  — 

—  230  — 

— 126  + 

—  127  + 

—  128  + 

—  129  + 

—  130  + 

+    54  + 

+    53  + 

+    52  + 

+    51  + 

+    50  + 

388 


fir     :! 
■  ^     :' 


TABLE  OF   PRODUCTS  OF   SINES  AND   COSINES. 


+    41°+ 

+    42°+ 

+  43°+ 

+  44°+ 

+  4B°+ 

+  139  — 

+  138  — 

+  137  — 

+  136  — 

+  135  — 

221  — 

—222  — 

—  223  — 

—224  — 

—225  — 

—  819  + 

—  318  + 

—  817  + 

—  318  + 

—  315  + 

sin 

C03 

sill 

cos 

sill 

cos 

sin 

cos 

sill 

cos 

0.1 

0.07 

0.08 

0.07 

0.07 

0.07 

0.07 

0.07 

0.07 

0.07 

0.07 

0.1 

0.2 

0.13 

0.15 

0.13 

0.15 

0.14 

0.15 

0.14 

0.14 

0.14 

0.14 

0.2 

0.3 

0.20 

0.23 

0.20 

0.22 

0.20 

0.22 

0.21 

0.22 

0.21 

0.21 

0.3 

0.4 

0.20 

0.30 

0.27 

0.30 

0.27 

0.29 

0.28 

0.29 

0.28 

0.28 

0.4 

0.5 

0:33 

0.38 

0.33 

0.37 

0.34 

0.37 

0.35 

0.30 

0.35 

0.35 

0.5 

O.fi 

0.39 

0.45 

0.40 

0.45 

0.41 

0.44 

0.42 

0.43 

0.42 

0.42 

0.6 

0.7 

0.46 

0.53 

0.47 

0.52 

0.48 

0.51 

0.49 

0.50 

0.49 

0.49 

0.7 

0.8 

0.52 

0.00 

0.54 

0.59 

0.55 

0.59 

0.50 

0.58 

0.57 

0.57 

0.8 

0.9 

0.59 

0.68 

0  00 

0.67 

0.01 

0.06 

0.63 

0.65 

0.04 

0.64 

0.9 

1.0 

O.fifi 

0.75 

0.67 

0.74 

0.08 

0.73 

0.69 

0.72 

0.71 

0.71 

1.0 

2.0 

1.31 

1.51 

1.34 

1.49 

1.30 

1.40 

1..39 

1.44 

1.41 

1.41 

2.0 

3.0 

1.97 

2.26 

2.01 

2.23 

2.05 

2.19 

2.08 

2.16 

2.12 

2.12 

3.0 

4.0 

2.C2 

3.02 

2.08 

2.97 

2.73 

2.93 

2.78 

2.88 

2.83 

2.83 

4.0 

5.0 

3.28 

3.77 

3.35 

3.72 

3.41 

3.00 

3.47 

3.60 

3.54 

3.54 

5.0 

fi.O 

3.94 

4.53 

4.01 

4.46 

4.09 

4.39 

4.17 

4.32 

4.24 

4.24 

6.0 

7.0 

4.5'.) 

5.28 

4.08 

5.20 

4.77 

5.12 

4.80 

5.04 

4.95 

4.95 

7.0 

8.0 

5.25 

6.04 

5.35 

5.95 

5.40 

5.85 

5.56 

6.75 

5.60 

5.66 

8.0 

9.0 

5.90 

0.79 

0.02 

0.69 

6.14 

6.58 

6.25 

(;.47 

0.30 

0.36 

9.0 

10.0 

6.50 

7.55 

6.09 

7.43 

6.82 

7.31 

6.95 

7.19 

7.07 

7.07 

10.0 

11.0 

7.22 

8.30 

7.36 

8.17 

7.50 

8.04 

7.64 

7.91 

7.78 

7.78 

11.0 

12.0 

7.87 

9.06 

8.03 

8.92 

8.18 

8.78 

8.34 

8.63 

8.49 

8.49 

12.0 

l:i.0 

8.53 

9.81 

8.70 

9.60 

8.87 

9.51 

9.03 

9.35 

9.19 

9.19 

13.0 

14.0 

9.18 

10.57 

9.37 

10.40 

9.55 

10.24 

9.73 

10.07 

9.90 

9.90 

14.0 

l.i.O 

9.84 

11.32 

10.04 

11.15 

10.23 

10.97 

10.42 

10.79 

10.61 

10.61 

15.0 

1(!.0 

10.50 

12.08 

10.71 

11.89 

10.91 

11.70 

11.11 

11.51 

11.31 

11.31 

16.0 

17.0 

11.15 

12.83 

11. 3S 

12.03 

11.59 

12.43 

11.81 

12.23 

12.02 

12.02 

17.0 

18.0 

11.81 

13.58 

12.04 

13.38 

12.28 

13.16 

12.50 

12.95 

12.73 

12.73 

18.0 

19.0 

12.47 

14.34 

12.71 

14.12 

12.90 

13.90 

13.20 

13.67 

13.44 

13.44 

19.0 

20.0 

13.12 

15.09 

13.38 

14.86 

13.04 

14.03 

13.89 

14.39 

14.14 

14.14 

20.0 

21.0 

13  78 

15.85 

14.05 

15.01 

14.32 

15.30 

14.59 

15.11 

14.85 

14.85 

21.0 

22.0 

14.43 

10.60 

14.72 

10.35 

15.00 

16.09 

15.28 

15., S3 

15.50 

15.56 

22.0 

23.0 

15.09 

17.36 

15.39 

17.09 

15.09 

10.82 

15.98 

16.54 

10.20 

16.26 

23.0 

24.0 

15  75 

18.11 

16.06 

17.84 

16.37 

17.55 

16.67 

17.20 

10.97 

16.97 

24.0 

25.0 

16.40 

18.87 

16.73 

18.58 

17.05 

18.28 

17.37 

17.98 

17.08 

17.68 

25.0 

2fi.0 

17.06 

19.62 

17.40 

19.32 

17.73 

19.02 

18.00 

18.70 

18.38 

IS.  38 

20.0 

27.0 

17.71 

20.38 

18.07 

20.06 

18.41 

19.75 

18.70 

19.42 

19.00 

19.09 

27.0 

28. 0 

18.37 

21.13 

18.74 

20.81 

19.10 

20.48 

19.45 

20.14 

19. SO 

19.80 

28.0 

29.0 

19.03 

21.89 

19.40 

21.55 

19.78 

21.21 

20.15 

20.80 

20.51 

20.51 

29.0 

30.0 

19.08 

22.64 

20.07 

22.29 

20.40 

21.94 

20.84 

21.58 

21.21 

21.21 

80.0 

cos 

sin 

cos 

sin 

cos 

sin 

cos 

Kin 

cos 

sin 

+  3] 

LI  — 

+  31 

2  — 

+  313  — 

4  31 

4  — 

+  3] 

5  — 

—  22 

3  — 

—  22 

8  — 

—  227  — 

—  22 

6  — 

—  2S 

5  — 

—  15 

J1  + 

-13 

2  + 

—  133  + 

—  13 

4  + 

—  13 

5  + 

+    4 

Q  + 

+   4 

8  + 

+    47  + 

+    4 

6  + 

+    4 

5  + 

